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```NONPARAMETRIC COMPARISON OF SLOPES
OF REGRESSION LINES

By

RAYMOND RICHARD DALEY

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA
1984

■'•-.■= •;<r;->fl'-»

To my parents

ACKNOWLEDGEMENTS

Dr. P.V. Rao suggested the research topic and, as my
major professor, provided many hours of consultation with me
during the preparation of this dissertation. For this I
thank him sincerely. I also thank the other members of my
committee, the Department of Statistics, and the staff of
the Biostatistics Unit. I am grateful to Dr. Anthony Conti
for allowing me flexibility in my work schedule that facil-
itated completion of this dissertation. I will not forget
Alice Martin for her practical advice and heartfelt support
during the various qualifying exams.

I thank my close friends for supporting my choice of
academic goals v/hile showing me the benefits of a balanced
life. I especially thank Christopher Kenward for his
encouragement and positive attitude. Finally, I appreciate
the love and support of my parents in all areas of m.y life
and I am proud to share the attainment of this long-awaited
goal with them.

..4^_j»5>„— -fl

Page

ACKNOWLEDGEMENTS iii

ABSTRACT V

CHAPTER

ONE INTRODUCTION 1

1.1 The Basic Problem 1

1 . 2 Literature Review 3

1.3 Objectives and Overview 9

TV70 COMPARING THE SLOPES OF TWO LINES 11

2.1 Introduction 11

2.2 Asymptotic Distributions 16

2.3 Large Sample Inference 32

2.4 Asymptotic Relative Efficiencies ... 35

2.5 Small Sample Inference 68

2.6 Monte Carlo Results 75

THREE COMPARING THE SLOPES OF SEVERAL LINES ... 99

3.1 Introduction 99

3.2 Asymptotic Theory and a

Proposed Test 104

3.3 An Exact Test 138

3.4 A Competitor to the Proposed Test . . . 141

3.5 Asymptotic Relative Efficiencies . . . 161

FOUR CONCLUSIONS 171

BIBLIOGRAPHY 175

BIOGRAPHICAL SKETCH 178

vV

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

NONPARAMETRIC COMPARISON OF SLOPES
OF REGRESSION LINES

By

Raymond Richard Daley

December, 19 84

Chairman: Dr. Pejaver V. Rao
Major Department: Statistics

Distribution-free confidence intervals and point
estimates are defined for the difference of the slope
parameters in a linear regression setting with two lines,
assuming common regression constants are used for both
lines. A statistic, T, is proposed for this setting that is
an application of a statistic discussed by Sievers in his
paper titled "Weighted Rank Statistics for Simple Linear
Regression," appearing on pages 628-631 of The Journal of
the American Statistical Association in 1978.

The statistic T employs weights, a , that are chosen

by the user. If the regression constants are x^ , . . . ,x ,

then under the weights a = x -x , the null distribution of

rs G r

T depends on the regression constants. An. associated exact
confidence interval for the slope difference can be obtained
by calculation of a permutation distribution, requiring use

of a computer. Under the weights a = 1 if x -x is

^ rs s r

positive, zero otherwise, the null distribution of T is
essentially that of Kendall's tau. In this case, an associ-
ated exact confidence interval for the slope difference can
be calculated using readily available critical values of
this distribution.

Simulation results indicate the power of a test of
parallelism based on T under either of the two sets of
weights dominates the power of other available exact proce-
dures. Pitman asymptotic relative efficiencies under equal
spacing of the regression constants also favor a test based
on T over other exact tests. Under unequal spacing of the
regression constants, the simulation results suggest use of

the weights a = x -x , when feasible.
^ rs s r

The method applied in the case of tv/o regression lines
is generalized to construct a test for use in comparing the
slopes of several regression lines with the slope of a
standard line. The proposed test statistic is a quadratic
form in a set of statistics, each having the structure of T.
The asymptotic relative efficiency of the statistic with
respect to a potential competitor is examined.

vt'

CHAPTER ONE
INTRODUCTION

o

1.1 The Basic Problem
There are many experimental settings where the appro-
priate statistical analysis involves comparing the slopes of
regression lines. For an example, consider a dilution assay
in which two different drugs are being compared to get a
measure of relative potency. A linear regression of the
response on the chosen dosage levels is constructed for each
drug. A fundamental assumption of this type of assay is the
assumption that the two regression lines are parallel
(Finney, 1964, p. 108). Hence a statistical test of paral-
lelism of two regression lines would be desirable in this
type of study.

In other applications, an estimate of the difference in
slopes might be needed. Consider a study comparing the
effectiveness of two different fluoride dentifrices in
reducing dental decay, where the participants engaged in
varying numbers of supervised brushing sessions. The
response measured is the change in the number of decayed and
filled surfaces over the duration of the study, called the
DFS increment. For each fluoride, the linear regression of
DFS increment on number of supervised brushings usually

results in a negative slope; as the number of supervised
brushings increases, the average DFS increment tends to
decrease. In this case the dental researcher might be
interested in an estimate of the difference between the rate
of decrease for the two fluoride groups, that is, an esti-
mate of the difference of the slopes of the two regression
lines.

We have just offered two examples of the basic problem
discussed in this dissertation, comparing the slopes of two
or more regression lines. In this work we deal with linear
regression models in the designed experiment, where the
levels of the independent variable are determined by the
experimenter, such as dose in a dilution assay. The methods
of inference we propose assume the levels of the independent
variable are chosen to be the same for each regression line,
as is frequently the case in a designed experiment. Thus,
in the dilution assay example, the chosen dosage levels for
both drugs would be the same.

Before proceeding further, we might mention that the
problem of comparison of slopes does not suffer from lack of
attention. As will be clear from the literature review in
Section 1.2, there are many methods for comparing the slopes
of two or more regression lines. The classical methods
based on least squares theory are exact when the underlying
distribution is normal and these methods are asymptotically
distribution-free over the class of distributions having
finite positive variance. Nonparametric methods exist which

are either distribution-free or asymptotically distribution-
free over a larger class of underlying error distributions,
including heavily tailed ones such as the Cauchy distribu-
tion under which the least squares methods perform poorly.
The nonparametric, asymptotically distribution-free methods
usually have good efficiency properties. Unfortunately, not
only is their performance for small sample sizes suspect,
but these methods are hard to implement in practice. In
this work, we concentrate on the problem of developing
better distribution-free methods of inference about slope
differences in situations where the experimenter has control
over the selection of the levels of the independent vari-
able.

1.2 Literature Review
In this section we will review articles in the litera-
ture discussing nonparametric analysis of linear regression
models. We focus on works that are pertinent to the topic
of this dissertation, comparing the slopes of two or more
regression lines. Often a nonparametric procedure designed
to make inference about the slope of a single line may be
adapted to the case of several lines. In Chapter Two we
will indicate a simple method of adapting a test designed
for the single line setting to the two line setting, when
the levels of the independent variable for both lines are
chosen to be the same. For these reasons we include a

discussion of the techniques applicable in the simple
(single line) linear regression setting.

To begin the discussion, consider the simple linear
regression model,

Yj = a + exj + Ej, (1.2.1)

j = 1, .,., N. The y's are the observed responses, a and B
are unknown parameters, the x's are the known regression
constants (levels of the independent variable), and the E's
are unobservable random errors. In this setting, nonpara-
metric tests based on rank statistics have been examined by
many authors (Hoeffding, 1951; Terry, 1952). Hajek (1962)
discussed the general properties of tests based on the
linear rank statistics.

N

V = E (X -x)<j,(R ), (1.2.2)

j=l J 3

where R^ is the rank of Y^ among Y^ , Y^ , ..., Y^^, 9 is a

score function which transforms the ranks to scores in an

N
appropriate way, and x = Z x./N. Hajek discussed

j=l ^

asymptotic properties of these tests which depend on the
chosen score function. Good discussions of properties of
the class of linear rank statistics (1.2.2) are found in
Hajek and Sidak (1967) and Randies and Wolfe (1979) .

Tests of the hypothesis B = based on the linear rank
statistics (1.2.2) can be used to derive estimates of g in
the model (1.2.1) by the application of a general technique
developed by Hodges and Lehmann (1963) . Since its appear-
ance in the literature, this technique has been applied to
estimation problems in a wide variety of settings. Because
Hodges-Lehmann estimates are used frequently in this disser-
tation, we now give a brief example describing this estima-
tion technique.

Suppose we assiime the simple linear model (1.2.1) but
with zero intercept,

Yj = exj + E., (1.2.3)

j = 1, ..., N, and we use the statistic (the numerator of
the Pearson product moment correlation)

N

Q = Z (X -x)Y. (1.2.4)

j=l J -*

to test B = against the alternative 6 5^ . Suppose
further that the errors E , j = 1, ..., n, are symmetrically

N
Q(6) = Z (x.-x) (Y.-Bx.)

j=l J 3 3

n'n'»)f,>S«?-»«

N
= Z (x^-x)E^ (1.2.5)

has median zero. Thus a desirable property for an estimate
B of 6 is that Q(3) be as near as possible to the median of
the distribution of Q(3), that is, as near as possible to
zero. Then the Hodges-Lehmann estimate of B based on Q is

the value of B such that Q(B) = inf |q(B) |. Of course this

B

example is in a special setting, but it does provide the
basic motivation for a Hodges-Lehmann estimate.

Adichie (1967) applied the technique of Hodges and
Lehmann to derive estimates of the slope and intercept in
the simple linear regression setting (1.2.1) using linear
rank statistics of the form (1.2.2). Jureckova (1971)
generalized these estimates to the multiple regression
setting. An undesirable feature of the methods of Adichie
and Jureckova is that calculation of the estimates and
confidence intervals requires iterative procedures.

Sen (1968) gave a complete development of an estimate
of slope in the simple linear regression setting (1.2.1),
that had earlier been suggested by Theil (1950) . The
estimate is the median of the slopes

Y - Y

^ . x< x^ , (1.2.6)

v -,

X r s

which is easy to calculate and intuitively appealing. The
Theil-Sen estimate can be derived by applying the Hodges-
Lehmann technique to the test statistic

Z2 sgn(x -X )sgn(Y -Y ) , (1.2.7)

r<s ^ ^ ^ ^

where sgn (u) = 1, 0, -1 as u is greater than, equal to, or
less than zero. The statistic (1.2.7) is the numerator of a
statistic known as Kendall's (1955) tau, a measure of
association between the pairs (x., Y.). Since the null
(3=0) distribution of Kendall's tau is distribution-free and
has been tabulated, and the statistic (1.2.7) has an easily
invertible form, exact small-sample confidence intervals for
6 are readily constructed using the Hodges-Lehmann technique
applied to the method of Theil and Sen; no iterative proce-
dure is necessary.

Scholz (1977) and Sievers (1978) extended the method of
Theil and Sen by using a weighted Kendall's tau. The
definition of a statistic equivalent to the one defined by
Sievers will be given in Chapter Two. Sievers also showed
how to construct confidence intervals and point estimates
for the slope 3 . Note that although Scholz ' s work appeared
prior to Sievers, the Scholz reference is only an abstract
of an unpublished paper and hence we hereafter refer to this
technique as the Sievers-Scholz approach. Further
discussion of the Sievers-Scholz approach will be given in
Chapter Two .

Having considered nonparametric methods of inference
about the slope in the simple linear regression setting, we
now consider those methods applicable to the multiple
regression setting. Recall the method of Jureckova (1971) ,
which was an extension of Adichie ' s (1967) results to the
multiple regression case. A test of the parallelism of
k >_ 2 lines using Jureckova 's method requires the computa-
tion of k individual Hodges-Lehmann estimates, each calcu-
lated by an iterative technique. Sen (1969) and Adichie
(1974) specify tests of parallelism of k ^ 2 lines that
require only a single Hodges-Lehmann estimate of the overall
slope, arrived at iteratively. The Sen and Adichie method
has good efficiency properties. However neither Sen and
Adichie 's approach, nor the technique of Jureckova, provides
exact confidence intervals for the individual slopes.

Jaeckel (1972) suggested estimating regression param-
eters by minimizing a chosen dispersion function of the
residuals. He showed that for a certain class of dispersion
functions his estimates are asymptotically equivalent to
Jureckova' s, but easier to compute. However, Jaeckel 's
estimates are also only asymptotically distribution-free and
require iterative computations.

Of the methods reviewed, only the Theil-Sen and
Sievers-Scholz statistics, applicable in the simple linear
regression setting, have readily invertible forms enabling
easy computation of exact distribution-free confidence
intervals for the slope parameter. In the two line setting.

.^i-,* "jy >cm M vfj-m^i^ f ^- HM— wwFmp ' ■■ " i f m i *y^f\\»r'» -^ >f , »■ -^ «Tr* tv-='*r"pTr>J>*- ■

two exact procedures are found in the literature. The
first, due to Hollander (1970) , is based on a statistic in
the form of the Wilcoxon signed rank statistic. The second
exact procedure, due to Rao and Gore (1981) , is applicable
when the regression constants of the two lines are equally
spaced. The statistic of this second procedure takes the
form of a Wilcoxon-Mann-Whitney two-sample statistic. These
two procedures are used to test the null hypothesis that the
two regression lines are parallel. The null distributions
of the two statistics are distribution-free, and exact
distribution-free confidence interals for the difference in
slopes can be readily computed by applying the Hodges-
Lehmann technique. More discussion about the Hollander and
the Rao-Gore methods will be given in Chapter Two.

1.3 Objectives and Overview
Having reviewed the literature regarding nonparametric
comparisons of slopes of regression lines, we now state
concisely the two objectives of this dissertation. The
first objective is to develop efficient, exact nonparametric
methods for comparing the slopes of two regression lines
when the researcher has control over the choice of the
levels of the independent variable. The methods we propose
will enable construction of exact distribution-free confi-
dence intervals for the difference between the two slopes.
The exact techniques of Hollander (1970) and Rao and Gore
(1981) , discussed briefly in the previous section, will be

10

direct competitors of the methods we suggest. A comparison
of these three techniques in Chapter Two using their Pitman
asymptotic relative efficiencies and a simulation study will
establish the superiority of the new methods whenever they '
are appropriate.

The second objective of this work is to generalize the
new methods suggested in Chapter Two to the setting where
the slopes of several regression lines are being compared.
In Chapter Three we will extend these new methods to the
multiple line case when the purpose is to compare the slope
of one of k lines, considered a standard or control, to the
slopes of the k-1 other lines. A comparison of our proposed
test to a modification of Sen's (1969) test will show that
our proposed test, in- addition to allowing an exact
distribution- free test for small samples not available with
Sen's approach, is almost as efficient as the modification
of Sen's test when the sample size is large and the error
distribution is not too heavily tailed.

CHAPTER TWO
COMPARING THE SLOPES OF TWO LINES

2.1 Introduction

In this chapter we examine the case of two regression
lines, assuming the same regression constants are used for
both lines. In this section we first establish the notation
for the linear regression model and the statistic used in
this chapter. We then motivate the form of the statistic
and give some background concerning its development. A
special characterization of the Hodges-Lehmann estimate
associated with the statistic is given. We close this
section with a brief look at the contents of the rest of
Chapter Two.

Consider the linear regression model

Y. . = a. + 6 .X . + E. . , (2.1.1)

i=l,2, j = l,...,N, and x- _< x^ £ . . . £ x . In this model a,,
a , B-, and g^ ^^® unknown regression parameters, the x's
are known regression constants, and the E's are mutually-
independent, unobservable random errors. We are interested
here in making inference about the slope difference.

11

12

6-,_2 = B-, - &2' Since the regression constants are assumed
the sanie for both lines, we suggest the use of the
Sievers-Scholz approach introduced in Section 1.2 when
discussing techniques appropriate in the simple linear
regression setting, now applied to the differences Z . = Y,

Ij
considered by Sievers and Scholz has the representation

- '^2i' J=lf''wN. A statistic equivalent to the one

T*(b) = {1/N)i:z a^^sgn(Z^-Z^-bx^_^) , (2.1.2)

r<s s r rs

where x = x - x , sgn (u) = -1,0,1 as u is less than,

^ o o X^

equal to, or greater than zero, and a > are arbitrary

rs — ■'

weights with a = if x = x .
^ rs r s

Note that under (2.1.1), the differences Z., j=l, ,N,

^j = ^1-2 "■ ^l-2^j -^ El-2,j' (2.1.3)

where a^_2 = a^ - a^, e^_2 = 6^ - 63' ^^^ E^_2^ . = E^. -
E_., j=l,...,N. In computing the differences Z., j=l,...,N,
we have reduced the two line case to the simple linear
regression case, enabling us to apply the approach of
Sievers and Scholz to the two line case. Of course the
assumption of common regression constants is crucial to this
reduction.

Let us now motivate the form, of T* (b) by first
discussing a special case of (2.1.2) due to Theil and Sen.

13

* *

Let Z . (b) = Z. - bx. and consider the pairs (x., Z . (b) ) ,

j=l,...,N. Under H : g = b, the value of x. has no

*
effect on the value of Z . (b) = a^_„ + E, _ .. However under

H, : Pi_2 > tjf a larger x. will tend to result in a

*
relatively larger observed Z . (b) = a,_„ + (3^ --b)x. +

^1-2 i* '^^^^ ^ method of constructing a statistic for

testing H- : &-,_2 = b is to employ some measure of associa-

*
tion between the x. and the Z . (b) , j=l,...,N.

Theil and Sen used this approach, selecting Kendall's

tau as the measure of association:

II sgn(x )sgn(Z*(b)-Z*(b))
r<s rs s r

[II sgn(x )ZZ sgn(Z^ (b)-Z^ fb))] ^
t<u ^" v<w ^

ZZ sgn(x )sgn{Z -Z -bx )
r<s s r rs

* U k ' (2.1.4)

*

Here N is the number of positive differences x - x ,

s r'

1 <_ r < s <_ N (keep in mind x^ _< x„ _< . . . _< x ) ,

N
i^) = N(N-l)/2, and we assume no ties among the Z's. The

*

. and Z .
: 3

statistic U-^(b) is Kendall's tau between the x. and Z . (b) ,

j=l,...,N, for each fixed value of b.

Let S = (Z -Z^)/x^^, 1 < r < s < N, denote the slope

of the line connecting the observed differences Z and Z .

r s

Since Z^ - Z^ - bx_^ > if and only if S > b, we see that
s r rs rs

Uj^, (b) is a function of these slopes. The numerator of U (b)
is equal to the difference between the number of slopes.

14

S^ , that exceed b and the number of these slopes which are
less than b.

Comparing U (b) given by (2.1.4) with T (b) given by
(2.1.2), we see the Sievers-Scholz statistic is an extension
of the statistic due to Theil and Sen obtained by replacing
sgn(x^ ) with a general weighting function a . This allows
the slopes determined from points which are farther apart to
be given more weight than the slopes determined from closer
points. Under fairly general regularity conditions, Sievers
has shown the highest efficacy of a test based on his
statistic is attained when the slopes S are given weights
a = X - X = X . Rewriting T (b) using the optimal
weights, a = x , we define

J. o XT S

T(b) = (1/N)ZZ x^^sgn(Z^-Z^-bx^^) . (2.1.5)

r<s s r rs

In the remainder of this chapter, we explore the appropriate-
ness of T(b) for inference about 6._2-

We have seen the Sievers-Scholz statistic is a
generalization of the statistic due to Theil and Sen. There
is a similar relationship between Hodges-Lehmann estimates
of the slope parameter associated with the Theil-Sen and
the Sievers-Scholz statistics. A Hodges-Lehmann estimate of
^1-2 ^^^^"^ °^ ^^^ Theil-Sen statistic can be shown to be
equal to the median of the set of slopes

{S^^: l_<r<s^N, X ^ x } (Sen, 1968) . The corres-
ponding Hodges-Lehmann estimate associated with the

15

Sievers-Scholz statistic T(b) is a generalization of the
Theil-Sen estimate and can be viewed {Sievers, 1978) as a
median of the distribution of a random variable V, where

P{V=v} = x^g/x.. if v=S^g, (2.1.6)

X.. = EZ X , l_<t<u_<N, Xt^x. Thus the Sievers-
t<u t: u

Scholz estimate is a weighted median of the slope estimates

{S^^: l<r<s<N, X t^x}.
rs — — r s

In the next section we briefly summarize Sievers'
results concerning the asymptotic distribution of the
Sievers-Scholz test statistic and estimate, now applied to
the two regression line setting. Using these results, we
describe large sample inference of e._2 in Section 2.3.
Pitman asymptotic relative efficiencies (PAREs) of the
Sievers-Scholz procedure with respect to other nonparametric
approaches as well as the least squares procedure are given
in Section 2.4. These PAREs are derived assuming equally
spaced regression constants. In Section 2.5 we propose
two exact tests of H^ : Q^_^ = which are easily implemented
for small sample sizes. Finally, we close Chapter Two with
a Monte Carlo study in Section 2.6. The first part of this
study concentrates on comparisons of the Sievers-Scholz
asymptotic procedure with others under moderately large
samples, while the second deals with exact tests and small
samples. Because PARE ' s are available only when the x's are
equally spaced, these Monte Carlo simulations emphasize

16

comparisons of the various test procedures under unequally-
spaced regression constants.

2.2 Asymptotic Distributions
In this section we present some important results
concerning the asymptotic distributions of the Sievers-
Scholz statistic and estimate. All of these results follow
from straightforward modifications of Sievers' (1978)
results for the simple linear regression setting. Sievers
presents his theorems without proofs, giving reference
primarily to the text by Hajek and Sidak (1967) . We repeat
these results, now applying them to the two line setting.
We indicate references to proofs or supply the basic steps,
since some of the results in this section will be needed in
Chapter Three when considering several regression lines.

As mentioned in the previous section, we are assuming
the optimal weights, a = x^ - x^, r < s, and thus we state
the results for T(b) in (2.1.5). Note that under these
optimal weights, T(b) can be expressed in terms of the ranks
of the differences Z . (b) , j=l,...,N, as follows:

N

*

T(b) = (2/N) Z [Rank(Z. (b))x.] - (N+l)5, (2.2.1)

j=l ^ 3

where Rank (Z . (b) ) is the rank of Z . (b) among {Z*(b):

N
r = 1,...,N} and x = Z x./N. Hence, T(b) is a linear rank

j = l -•

17

statistic and distributional theory for this class of
statistics applies to T (b) . For example, assuming some
regularity conditions^ asymptotic normality of T(0) under
Hq: e-]^_2 = is immediate (see Theorem 2.2.1).

Basic notation . We now give some notation that will be
maintained throughout Chapters Two and Three. Let

2 N
a = Z (x_.-x)^/N,
^ j = l ^

(2.2.2)

^T,A = ^(^j-^)^/3 = Na2/3,

(2.2.3)

s-1

s j^;^ s :

(2.2.4)

X

N

Z

j=r+l

(x.-x ) , and so

(2.2.5)

x.. - X . . = N(x.-x) and

(2.2.6)

2 2^-2

ZZ X = EZ (X -X )^ = N Z (x.-x)^,

r<s ^^ r<s ^ ^ j=l ^

(2.2.7)

by simple algebra. For ease in notation, we assume the
underlying error distributions of the two lines are the
same, with cumulative distribution function (cdf) F and
probability density function (pdf) f. Let E^ , j==l,2,...

■ff U ' HJjJl 'ii ii * w ^ . I 1 111 i »jn i j^j ii un.; ! u t[iiiy*wiwrywwiiig?WM'i"*'^a<>EWI Gil l JI, *M U p

18

designate independent and identically distributed random
variables with cdf F and pdf f; G and g will denote,
respectively, the cdf and pdf of E^ - E^ . The cdf and pdf
°^ ■^l ~ ^2 ~ -^3 ^ ^4 ^^11 ^^ denoted by H and h,
respectively. For general pdf g, let I (q) = fq^'{x)dx,
provided the integral is finite. The normal distribution
with mean y and variance a^ will be designated
by the notation N{y,a ). We adopt the 0(O, o{'),
°p^'^' °p^*^' ^^'^ '^ notations as in Serfling (1980, p. 1,
8-9) . Let P indicate convergence in probabilitv and ^
indicate convergence in distribution. Finally, let a(b+c)
be notation for the interval ( | ab | - | ac | , | ab | + | ac | ) .

Conditions. The following conditions are used in the
statements of the theorems of this section:

N

_ 2 2

2 (x.-x) = Na -)■ 00 as N ->■ »

j = l ^ , ^ CIO iN .

II. max (x.-x) max (x.-x)^

l^J^N -^ l<;i^N -

N 2 ^ ° as N ■>

Z (X -5)2 Na^

j = l ^

III. G is continuous.

IV. G has a square integrable absolutely continuous
density g.

Condition II, the familiar Noether (1949) condition for
regression constants, ensures that no individual regression
constant dominates the others in size as N ^ -

19

Theorem 2.2.1 ; Under H^ : ^^_^ = b and condition III,
the exact variance of T (b) is [ (1/3 ) + (1/N) ] Na,^ . If
^0" ^1-2 " ^ ^^'^ conditions 1, li, and III hold, then

^^^)/''t,A - N^°'l) (2.2.8)

as N ->■ «.

Proof;

From the definition of the Z.'s (2.1.3),

's - ^ - ^^rs = ^1-2, s - ^1-2, r + <^l-2-^)^rs- <2.2.9)

Clearly then, the distribution of T (b) (2.1.5) when 6, = b
is the same as the distribution of T(0) when 6-, - = 0. So
assuming ^-^_2 = 0, we derive the variance of T(0).

Var[T(0)] = Var[(l/N) II x sgn (E, , - E, ^ ) 1

r<s rs ^ 1-2, s 1-2, r'^

* ' r't's''^^"*^ =°''[=9°<=l-2,s-^l-2,r'' =5n (Ei.2,s-=l-2 ,t' ^

" ' r's'u''-^"^" '^°"'=9'"^l-2,s-^l-2,r'' =9" '=1-2 ,u-=l-2 , = ' "
= (l/N^) {Cj + Cj + C3 + c^}, (2.2.10)

f

¥ * -- ' i--X' ^^g"g^g^|^gg^*W r^i5> "^; :jP!*Q»gJ g '«'**^M M P« *-L» B;i'' J w .i;» i Nai M« Ct .

20

where c^, c^, c^ , and c^ denote the 4 terms in this variance
expression. Since E ^ ^ - E is symmetric about zero,
it follows that E[sgn(E^_2 ^ - E^_^ ^) ] - o. Also since

'2_2 s ~ ^1-2 r ^^ continuous, it follows that

2
sgn (E.|^_2 g - E^_2 j.) = 1 with probability one. Thus

^1 = 11^ x's^^-f^^^ (^l-2,s-^l-2,r)^

11^ 4s ^f^^^' (^l-2,s-^l-2,r)^

22 X (2.2.11)

r<s ^^

To simplify C2 we first note

Gov [sgn (E^.s , g-E^.s ,^) , sgn (Ei_2 ^^-Ei_2 ^^) ]

= E[sgn (Ei_2,3-Ei_2,r) sgn (Ei_2 ,^-Ei_2 ,P ]
= t^ ^^1-2, s - ^1-2, r ' 0' ^l-2,u " ^1-2, r > '^
+ ^ ^^1-2, s - ^1-2, r < 0' ^1-2, u " ^1-2, r < 0>^

- f^ ^^1-2, s - ^1-2, r ^ °' El-2,u - ^1-2, r ^ 0>

^ ^ ^V2,s - ^1-2, r ^ °' ^l-2,u - ^1-2, r > °>^
= [ /[l-F(e)]^dF(e) + /F^(e)dF(e)]

- [ /[l-F(e) ]F(e)dF(e) + /F (e) [1-F (e) ] dF (e) ]■
= [(1/3) + (1/3)] - [2(1/6)]

1-2 s "^^ continuous (condition III) and consequently ;

21

- l/3f (2.2.12)

and so it follows that

'2 ^ \<s</rs-ru ^^^^ ^^^ ^^1-2 ,s-^l-2 ,r) '^^^ ^^1-2 ,u-^l-2 ,r) ^

(1/3) (2 ZZS X X )

rs ru
r<s<u

(1/3) (I xj ) (using (2.2.5)). (2.2.13)

r ^•

Similar basic manipulations yield the following
simplifications of c- and c . :

'3 = (1/3) (Z X h, (2.2.14)

r

c^ = - (1/3) (2 Z x^^x^^). (2.2.15)

Substituting these simplifications into (2.2.10) we have

Var [T(0)] = (l/N^) { ZZ x^

r<s ^^

+ (1/3) Z (x^^ + x^^ - 2x^_x^^)}

= (l/N^) {J:z xJ + (1/3) I (x^ -x )^}. (2.2.1&)
r<s ^^ r ^' '^

Using (2.2.6) and (2.2.7) in (2.2.16) yields

22

Var[T(0)] = (1/N^){N E (x.-x)"^ + (1/3)N^ I {x.-S)^}

j=l 3 j=i 3

^ - 9

= Z (x -x)^ { (1/N) + (1/3) }
j = l J

= [(1/3) + (1/N)] Na^, (2.2.17)

o

verifying the expression for the exact variance of T (b)
under H^ : B = b.

If we take k=2 in Theorem 3.2.3 then this theorem
states that under H^ : e^_2 = 0' assuming conditions I, II,
and III

T(0)/a^^^ J N(0,1) (2.2.18)

as N ^ 00. From the remarks at the beginning of this proof
T(b)/a^^^ has the same limiting distribution under H_ :
^1-2 ^ ^' ^^^ ^° ^^^ proof is complete. Note that 0^ is
the asymptotic variance of T(b) under H : 3 ^ = b.

Theorem 2.2.2 ; Assume a sequence of alternatives

J,
^N* ^1-2 "^ ^1-2^^^ "^ to/(N^a^) to the null hypothesis H- :

2l_2 = 0' where to is a constant. Then under conditions
I-IV,

'^^°^/''t,A t N(2(3'S)a)I(g),l) (2.2.19)

as N ->■ 00

-•»w ajj< > u*j. i i i' m i L i' J»i » iii i i.w i-ii P , III 1 . 1 m m rajnu ,« i M«

23

Proof ;

A detailed argument is given, since results obtained
here are used in the proof of Theorem 3.2.4 in Chap-
ter Three.

Let <^i^ = 0)/ (N^o^) . From (2.2.9) and the definition of
T(b) (2.1.5), it is clear that the distribution of T(0)
under e^_2 = im^ is the sam.e as the distribution of T(-a) )
under ^-^^2 ~ ^' Therefore the proof is complete if we
assume &2_-2 ^ ° ^^^ show that T (-cu ) has the desired
limiting distribution. V7e first state and prove two lemmas.

Lemma 2.2.3 : Assume 3^_ = and conditions I-IV.

Then

E[T (-03^)70^^^] ^ 2(3')a)I(g) (2.2.20)

as N -»■ «>.
Proof of lemma:

E[T(-a3^)] = (1/N)ZE x^^E [ sgn (Z^-Z^+co^^x^^^) ] .

r<s

Now

E[sgn(Zg-Z^+aj^x^g)] = 1 - 2/G (z-u^x^^) dG (z) . (2.2.21)

^>^>»i^>*v. ->r»V ^>V'>_:.i-^r»

24

Conditions III and IV allow us to write a Taylor's series

expansion of G(2-MX) : G(2-cux^) =G(z)-cax a(z-e (z)),

j-NXo Mrs Nrs" rs

where < |9^^(z)| £ ^N^^rs ' " ^^^""^

max X . -X

^N^rsl = U(x^-V/(n\H < 2 M i£J|N_i

N^ a

X

it follows from condition II that 6 (z) ^ uniformly in r

rs -'

and s as N -^ ". That is, for all e > 0, there exists N(e)
such that

N > N(e) => Icoj^x^gl < e for all r < s. (2.2.22)

Then by absolute continuity of g (condition IV) , for all
6 > 0, there exists N {&) such that

* •
N > N (5) => g(2-6^g(z)) e (g(2)±<S), (2.2.23)

for all 2. Since g(-) and 6 are nonnegative and g(«) is
square integrable (condition IV), multiplying by g(z) and
integrating we see

*
N > N (6) => /g(z-e^g(z))g(z)dz e (I(g)±6). (2.2.24)

Substituting the Taylor's series expansion into (2.2.21) we

have

E[sgn(Z^-Z^+o;j^x^^)] = 2a)^^x^^/g (z-e^^ (z) ) g (z) dz ,

-.rr- T^v-".;^!,!,.

25

and so

E[T(-a.^)/a^^^] = (2a)^/ (Na^ ^^) ) Z Z x^^/g (z-G^^ (z ) ) g (z ) dz

X ^ o

Assume tu > 0. Using (2.2.24) in the above representation we
see N > N (6) =>

E[T(-aj^)/a^ ^] e {2a)^/ (Na^ ^))IZ x^ (1(g) ±5)
' ' r<s

^ - 2
(2a)/(NG ))(N Z (X -x)^) (I(g)±6) (from (2.2.7))

' -1=1 J

2(3^)a)(I(g)±6) ,

and so, N > N* (6/ (2 (3 ') u) ) =>

E[T(-cj^)/a^^^]e(2(3^)ujI(g)±6) ,

and (2.2.20) is proved for w > . The proof is similar for
for 0) < 0.

Lemma 2.2.4 : Assume 6,_2 = and conditions I, II, and
III. Then

III f^\-*'\^ fm »-»^%r^J'imm'**t r'i:' mm mj m ft .jr

26

as N ^ " .
Proof of lemma :

Since under &^_^ = 0, E[T(-a)j^) - T(0)] - E[T(-aj^)],
we prove (2.2.25) by showing

Var[(T(-a)jj)-T(0))/a^^^] -^ (2.2.26)

as N ^ ». First, T(-a)j^) - T(0) = ^IZ ^rs^rs ^"n^ '

where E^^it,^^) = sgn (Z^-Z^+oj^x^^) - sgn(Z^-Z^).
Assuming co > ,

2 -tj„x < Z - Z <
otherwise.

We will suppress the argument ui^ of H^^ (u^^) . Then, since
'Vrs' "^ ° uniformly in r and s (see (2.2.22)) and the cdf

of 2g-Z^ is continuous (condition III) , it follows that

2

E[H^^] = 4P{-aj^x^^<Zg-Z^<0} ^ uniformly in r and s as

N > ". Similarly, E[H H , ,] > uniformly in r, s, r' ,

s' as N > <». So Var(H ), Cov(H ,H ), and other terms of

J- o r s r u

that form are all uniformly o(l) as N ■» ». Now

Var[(T(-a)^)-T(0))/a^^^]

(N^aJ ) ^Var[EZ x H ]
-' r<s ^^ ^^

lj*t*;:«|«e^»|,-i-«[j>

27

= (N2aJ^^)-l(ZE^x2^Var H^^

+ 2 ZEE X X Cov(H ,H )
rs ru rs ru
r<s<u

+ 2 ZEE x^^x, Cov(H ,H^ )
r<t<s ^^ ^^ ^^ t^

+ 2 EEE X X Cov(H ,H ) }
r<s<u ^^ =^ ^^ ^^

2 2 -1
" ^^ '^T,A^ (0^+02+03+0^), (2.2.27)

where c^, <z^, o^, and c^ denote the 4 terms in this varianoe

expression. Since r < s < u implies < x < x , it

^ — rs — ru'

follows that

C^ < 2 EEE X X |Cov(H ,H )
~ r<s<u ^^ ^^ ^^ ^^

< 2 EEE x^ ICov(H ,H )
— ru ' rs ru
r<s<u

= 2EEju-r-l)x2jcov(H^3,H^^)|.

Now (u-r-1) <_ N for all 1 £ r < u < N, and

c 1 < 2NEE x" |Cov(H ,H )
2' - ^^^ ru' ' rs' ru'

so

28

The two similar terms c_ and c. in (2.2.27) can also be
bounded in this v/ay. Combining these bounds with the fact
that all variance and covariance terms in (2.2.27) are
uniformly o(l) we find

Var[((T-(o^)-T(0))/a^^^] < B^,

where B^ = (o(l)/N^a^ ^) (6N+1)ZZ x^

' r<s

(3o(l)/N^a^) {6N+l)N^a^

= o (1) , as N -> ",

and hence (2.2.26) is proved for u > . The proof is
similar for u < .

Proof of Theorem 2.2.2 (continued) ;

We see by Lemma 2.2.3, Theorem 2.2.1, and Slutzky's
theorem (Randies and Wolfe, 1979, p. 424) that under
Hq: B^_2 = 0,

T(0) + E((T(-a) ))

^^— J N(2(3^)a)I(g) ,1)

T,A

as N -> «=. Using this result. Lemma 2.2.4 and another
application of Slutzky's theorem shovz that under

^0- ^1-2 " °'

cffc — To«a »g- iT^

29

T(-(^N^''''t,A t N(2(3^)wl(g) ,1)

as N ^ ". Then by our remarks at the beginning of the proof
of Theorem 2.2.2, we are done.

Consider the Hodges -Lehmann estimate of 6,_^, say 6, ,,
associated with the statistic T(b). If we let
^1_2 = sup{B: T(3) ^0} and i^_^ = inf{B: T(3) < 0}, then
we may define Q^_^ = (6^_2+eJ_2) /2 . We now give a theorem
concerning the asymptotic distribution of 3t_,.

Theorem 2.2.5 ; Under conditions I-IV,

^^^x(^l-2"^l-2^ f N(0,(12l2(g)) 1) (2.2.28)

as N ->■ ".

Proof :

■^®^ "n ~ '^Z (N'^^^) where w is a constant. Let P„{'}

h

denote the probability calculated when 3 = 3, and let <
designate the standard normal cdf. From Theorem 2.2.2 it
follows that

lim P {T(0)£0}

lim P {(T{0)/a ) - 2(3^)ajl(g) < -2 (3^) cal (g) }

^a j WJT«i » IJ il im i U^j yi . W r-7 <ali

30

(-2(3 )ul(g)), or equivalently ,

lim P_ {T(C)£0} = * (2 (3^)0)1 (a)) . (2.2.29)

By the definition of &^_^ and 3^_2 and using the fact that

T(b) is nonincreasing in b, we have

^1-2 ^ ^ "^ ^^^^ ^ ° "^ ^1-2 - ^' (2.2.30)

^1-2 < ^ => T(b) < => bJ_2 ^ b. (2.2.31)

Since by condition III the underlying distribution is
continuous, it can be shown (using a proof similar to that
of Theorem 6.1 of Hodges and Lehmann (1963)) that the
distributions of e^_2 and ^2,-2 ^-^^ continuous. Thus
(2.2.30) and (2.2.31) imply that

Pp ii^ <h} = P {T(b)<0}, (2.2.32)

P {8^ <b} =.P {T(b)<0}. (2.2.33)

^1-2 -^ ^ Pi_2

Since B^_2 = (1/2) (6^_2 + 65^_2) and 6^_2 < l^_2' it follows
that

^6,_2^^?-2<^> 1 ^B^_2^^1-2<^> 1 ^^_2^^1-2<^>'

! ■»;» " . ii LJ« «M''*qw»Mwwww«»'wM«»«g°w5gT«»»°B wa J i ^ M U-. ■ im i g.'i^m. iii jB.ir^T—'rj 'i mf * m n iiiiiii.i . i i im o— — bm

31

and then substituting (2.2.32) and (2.2.33) into this result
we have

P {T(b)<0} < P {6 <b} < P. {T(b)<0}. (2.2.34)

Because the distribution of T(B) when ^-,_2 = 3 is the same
as the distribution of T(0) when 6,_p = (see proof of
Theorem 2.2.1) it can be shown using the definition of B
that the distribution of (^i_2~^i_2^ under &^_^ is the same

A

as the distribution of &-,-2 ^^®" ^i_2 "^ ^' Since the dis-

A

tribution of 3-J.-2 ^^ continuous and the limiting distribu-
tion in (2.2.29) is continuous, we apply (2.2.34) to obtain

lim P {N'a (6 -3 ) < to}

= lim P-^{N^a^6T ^ <cj}

lim P {3 < u^/ {N^'o ) }

lim P {T(aj/(N'a^) ) < 0} (from (2.2.34))

= lim pQ{T(a3j^) < 0}

-J«IJJl.J!'.]JieWW'^WSlJ-.'B i! im.BlMmMW.UM ^i^1l gW<g - «• "■ - ' ■ »i i..r nTOi.iia i. »« j i mi--. ca»

32

= lira P {T(0) _< 0}

= §(2(3^) a)I(g)) (from (2.2.29)), (2.2.35)

and the result (2.2.28) follows immediately.

2.3 Large Sample Inference
Using the results of the previous section, we now
proceed to construct tests and confidence intervals for
^l_2- These tests and confidence intervals follow from
those presented by Sievers, now applied to the two line
setting.

Consider a test of H^ : B^_2 = &q against H^ : Q > 6

based on the statistic T(eQ). Of course to test for
parallelism of the regression lines we take g. = 0. Large
values of T(3q) indicate the alternative H^ holds. For a
given < a < 1, let z denote the 100 (1-a) percentile of
the N(0,1) distribution. Then from Theorem 2.2.1, the test
which rejects H if

■"6o'''='t,a' \

is an approximate level a test. A two-sided test of

H

'2__2 = is derived by making the usual modifications.

Since T (b) (2.1.5) is a nonincreasing step function of
b with jumps at the slopes S = (Z -Z ) /x , 1 < r < s < N,
a confidence interval for g can be constructed by

33

inverting the two-sided hypothesis test. This interval has

endpoints that are properly chosen values from the set of

slopes {S : 1 <_ r < s <_ N}.

Let J(s) be the cumulative distribution function of a

discrete random variable that takes the value S with

rs

probability x /x. . , where x.. = ZZ x . Then for each b

r<s

such that b7*S ,l_<r<s_<N,

J(b) = (1/2) [l-(NT(b)/x..)] . (2.3.1)

If the distribution of the Z ' s is continuous, then
P{S^g = b for some l_<r<s_<N} = for each b and (2.3.1) holds
with probability one. Therefore if we can determine
constants t^ and t^ such that P{t, <T (6) <t2 } = 1 - a, then

1 - a = P{^(l-(Nt2/x..)) < J(e) < J2(l-(Nt^/x..)) }

= p{j;;;-^(^(i-(Nt2/x..))) ^ e < jj-(h{i-{i>it^/x. .)))},

(2.3.2)

where J~ (u) = inf {s: J(s)>_u} and j"""" (u) = inf {s: J(s)>u}
are inverses of J defined appropriately for our purpose.
Thus

[J^-^(%(l-(Nt2/x..))) , J^^(35(l-(Nt^/x..)) )) (2.3.3)

--*-»»^»^«Wi-— .--T^CH';- -.^■.i-a''*>t:i>- »■■"--■ ~r<l.-JL. -..*■■■ T«JC— *».»f*'— ^-^

34

is a 100 (1-a) percent confidence interval for 6, ^.

If V7e use the asymptotic normality of T(3) in Theorem
2.2.1 to determine t^ and t^ we find that

^^~-^ ^'^-^a/2 <N^T,a/2^- • ) ) ' ^'J' ("s+z^/s (Na^^A/2x. . ) ) )

(2.3.4)

is an approximate 100 (1-a) percent confidence interval for
^1-2* ^^ ^^^ write this interval more explicitly in terms
of the slopes S^^ as [S^'^^s'^) , where

S = min{S^g: J (S^^) >%-2 , (Na /2x. . ) , l£r<s<N},

(2.3.5)

and

S = miniS^^: J {S^^)>h+z^^^{lio^ ^^/2x. .) , l<r<s<N}.

(2.3.6)

An alternative confidence interval follows directly
from the asymptotic normality of &^_^ (Theorem 2.2.5). An
approximate 100 (1-a) percent confidence interval for e._ is
given by

^1-2 ± 2,/2(2(3V)a^i(g))-l,

where 1(g) is a consistent estimate of 1(g). Such estimates
have been proposed by Lehmann (1963) and Sen (1966) .

35

2.4 Asymptotic Relative Efficiencies
In this section we compare the Sievers-Scholz approach
for testing H^ : e^_2 = b with the procedure due to Hollander
and the procedure due to Rao and Gore. As noted in Chap-
ter One, the Hollander and Rao-Gore procedures include exact
confidence intervals for the slope difference 5-,_2' We will
see in the next section that our proposed application of the
Sievers-Scholz approach also includes exact confidence
intervals for &2.-2' ^^^^^- these three exact, nonpararaetric
procedures are the primary focus of the asymptotic relative
efficiency comparisons presented here. Comparisons of the
Sievers-Scholz test with the classical t-test based on least
squares theory and Sen's (1969) test are also made. First
we show how to construct each of the competing nonparametric .
test statistics, along with a brief illustration of the
rationale behind each one. We then describe a sequence of
sample sizes tending to infinity for the purpose of
computing the Pitman asymptotic relative efficiencies
(pares) . Finally', we compute these efficiencies and compare
their values assuming several different underlying error
distributions.

Since all three exact, nonparametric procedures can be
expressed in terms of basic slope estimates for each line,
we first define

^irs = ^"is-^ir^/^rs' ^r " ^^s' (2.4.1)

nMMi^«P9f«

36

the estimate of the slope of line i (i=l,2) resulting from
the responses at x and x . Recall that in the previous
sections we used a similar notation.

3 = (Z -Z )/x ,
rs s r rs'

to designate slope (difference) estimates computed from the

differences Z., j=l,...,N. Of course S = S^ - S^ , and
3 rs Irs 2rs

the additional subscript indicating line 1 or 2 in the
estimates in (2.4.1) should help to avoid any confusion.
There will be {^^) = N(N-l)/2 of these estimates associated
with each line. The slope estimates of line 1 are naturally
independent of those of line 2, but the (2) slope estimates
of a single line are not mutually independent. One way of
motivating and com.paring the three exact, nonparam.etric
procedures is to examine how they utilize these basic slope
estimates in forming their test statistics.

The Sievers-Scholz Statistic

In the two line setting, the Sievers-Scholz statistic

is appropriate only when the lines have common regression

constants. As before, let x- < x„ < . . . < x„ denote the

1 — 2 — — N

regression constants. We can write the Sievers-Scholz
statistic T(0) in terms of the slope estimates:

T(0) =lEr x^3sgn(s,^.3-S2^^)
r<s

37

Examining this representation we see that each line 1 slope
estimate is compared v/ith the line 2 slope estim.ate
resulting from the observations at the same regression
constants. Thus all (p) slope estimates of each line are
used, but an estimate from line 1 is compared only with the
corresponding estimate from line 2. This results in („)
comparisons across lines. Each comparison is weighted by
the distance between the x's used in its construction.

The Hollander Statistic

Unlike the Sievers-Scholz statistic, the Hollander
statistic is applicable even when the lines do not share
common regression constants. However, use of the Hollander
statistic requires a grouping scheme designating N/2 pairs
of regression constants for each line. Assume N = 2k. When
the x's are equally spaced, that is, when the regression
constants for line i are

X. T = L. + mc, m = 0, 1, ..., 2k-l, (2.4.2)

i = 1, 2, for some constants L^ , L„, c^ , and c_, Hollander's
grouping scheme pairs

X. ^ with X. ^,, , m = 1, ..., k, (2.4.3)

i,m i,m+k

i = 1, 2. The first step in Hollander's procedure is to
utilize the observations at each pair of x-values to

38

construct N/2 independent slope estimates of the fom

(2-4.1) for each line. Hollander notes that under equal

spacing (2,4.2) his grouping scheme (2.4.3) minimizes the

variance of the slope estimates used among all grouping

schemes that produce identically distributed slope

estimates. Under unequal spacing of the regression

constants, some ambiguity exists as to the choice of a

grouping scheme. Hollander suggests devising a scheme that

will yield pairs of x-values situated approximately a

constant distance apart.

Having computed N/2 = k independent estimates of the

slope of line 1 and k independent estimates of the slope of

line 2, the next step in Hollander's procedure is to

randomly pair the slope estimates of line 1 with those of

line 2. Let (S-j^^.^, ^2^^) designate one of the k pairs. For

each pair the difference of the form S, - S^^ is

Irs .2tu

calculated. If we label these differences d^ , . . . ,± then

1 k

Hollander's statistic is computed as

W = ^ ^m^^'^m^' (2.4.4)

m=l

where R^, is the rank of |d^,| among {|d^|: m=l,...,k}, and
6 (a) = 1 if a > , otherwise; W is the Wilcoxon signed
rank statistic computed using the k slope difference
estimates as the observations. Writing a slope difference

^•ft w p y M gw pn y 1 1 'im \\ t» m mi-ykm^ imm^Qr»\mo»mv ?9 e'^'* y*l»'SK~mr r

39

estimate d^ in terms of the underlying parameters, we see d

has the form

where x^^^ = x^^ - x^^, i = 1, 2. Since E^^ and E^^ are
independent and identically distributed, it follows that the
distribution of E^^' - E^^ is symjr.etric about zero. Clearly
then ,

^is - ^ir ^2u - ^2t
""irs ^2tu

is symmetrically distributed about zero. Thus Hollander's
approach does not require the same regression constants or
equal spacing since the Wilcoxon distribution will apply
under 6^_2 = regardless of the spacing or choice of
regression constants. However, if the regression constants
are the same and equally spaced, the asymptotic relative
efficiency (ARE) to be presented in this section as Result
2.4.2 indicates superiority of the Sievers-Scholz approach.

Note that Hollander's approach does not use all
available (2) basic slope estimates from each line. Instead
a subset of N/2 independent line 1 slope estimates is
selected from the {^) possible. A similar set is selected
from the line 2 slope estimates. Each member of the first

40

set is compared with only one randomly selected member of
the second set, resulting in only N/2 comparisons of slope
estimates across lines. This is a sm.aller number of
comparisons across lines than the (2) comparisons of the
Sievers-Scholz method. Although there are dependencies
among the Sievers-Scholz comparisons, the greater number of
them would lead us to expect the Sievers-Scholz approach to
be superior to Hollander's method when both are applicable.
Again, the ARE Result 2,4.2 will confirm this.

The Rao-Gore Statistic

Let A(B) designate the set of N/2 slope estimates for
line 1(2) used in constructing Hollander's statistic. Note
that under equal spacing (2.4.2) with c^ = C2 = c and using
Hollander's recommended grouping scheme (2.4.3), the
distribution of the slope estimates in A differ from those
in B only by a shift in location of 3-i_2 "= ^i ~ ^2'

E, - E^
q - R + Is Ir
^Irs " ^1 ^ kc

c - B + ^2u - ^2t
^2tu ^2 ^ kc

Here x. , - x. = kc is the common distance between pairs
i,m+k i,m

of x's used in forming the slope estimates. Thus, under
equal spacing, Rao and Gore proposed the Mann-Whitney-
Wilcoxon statistic of the fcirm

41

" -{S.^JeA}{S2^^eB} ^ ^Slrs'^2tu)' (2.4.5)

where y{a,h) = 1 if a > b, otherwise. It is clear that
the Rao-Gore procedure would apply under any spacing and
grouping scheme resulting in the same distance between all
pairs of x's used to form slope estimates. Under such
spacing, the Rao-Gore procedure eliminates the extraneous
randomization needed by Hollander's procedure to pair the
slope estimates of the two lines.

We see the Rao-Gore procedure uses the same two sets of

slope estimates as Hollander's approach. However all

2
possible (N/2) comparisons across lines are made. This

leads us to expect that the Rao-Gore procedure will compete
favorably with Hollander's method.

Comparing the Rao-Gore technique with the Sievers-
Scholz approach in the previous intuitive way is not as
revealing. The Rao-Gore technique makes all possible
comparisons across lines among two relatively small sets of
independent slope estimates. The Sievers-Scholz approach
makes only pairwise comparisons across lines, but uses all
possible slope estimates. The sets of slope comparisons
used in the two procedures are such that neither set is a
subset of the other. We will see, however, that the ARE
Result 2.4.3 indicates superiority of the Sievers-Scholz
approach.

■ in.ii i an i i_». i .,

42

The Sen (1969) Statistic

Sen (1969) proposed a statistic for testing the
parallelism of several regression lines. We introduce his
statistic here in the two line setting. Note that Sen's
statistic does not require common regression constants for
the two lines as required by the Sievers-Scholz statistic.
However we will assume common regression constants to ease
the notation. Modifications needed to construct Sen's
statistic in the more general case of different x's for the
two lines will be obvious. Thus we assume the basic linear
model (2.1.1) .

Let (|) (u) be an absolutely continuous and nondecreasing
function of u: < u < 1, and assume that ((> (u) is square
integrable over (0,1). Let U^^^ < U^2) < ••• < U^^^ be the
order statistics of a sample of size N from a uniform [0,1]
distribution. Then define the following scores:

Ej = E[MU(j))], (2.4.6)

or

Ej = <j)(j/N+l), (2.4.7)

j = 1, , N. Define

1

** = / 0(u)du (2.4.8)

and

■■I.W .mimu g'ma^'Sf^

43

2^2 2

A = f ^ (u)du - (<i)*)^, (2.4.9)

and consider the statistics

V = [ E (x.-x)E ]/(An\ ), (2.4.10)

3=1 -^ ij

i = 1, 2, where R^^ is the rank of Y^. among Y., Y.^, ...,
^iN' ^^^ observations of the ith line. Statistics such as
(2.4.10) are used in the single line setting to test

The function (j) (u) is called a score function. It is a
function applied to the observed ranks of the Y's. The
choice of cj) can be made to achieve desirable power
properties of a test based on (2.4.10). These properties
depend in part on the underlying error distribution. A
general discussion of score functions is beyond the scope of
this work. We will discuss the score function here only to
the extent needed to clearly present Sen's statistic.

Assume F, the underlying error cdf of the assumed model
(2.1.1) , is absolutely continuous and

I*(F) = / [|^]2dF(x) < ». (2.4.11)

wp <i"'ji^wj: iT ^p-. g^ BRm a

44

Then we reserve the symbol ^ (u) for the following score
function:

,(u) = _ fMZ^iluU , < u < 1. (2.4.12)

f(F ^(u))

It can be shown.

1
/ 1'(u)du = 0, and

1 2
/ 'P (u)du = I*(F) .

The score function Y (u) has been shown to have certain
desirable properties when applied to the two-sample location
problem (Randies and Wolfe, 1979, p. 299).
We also define

1

Pi'^r<i>) = [/ 'i'(u)^(u)du]/[A^I* (F)]^, (2 4 13)

which can be regarded as a measure of the correlation
between the chosen score function <p and the optimal one C?)
for the error distribution being considered- The expres-
sions p(V,(j)) (2.4.10) and I* (F) (2.4.11) will appear in the
development of the statistic based on Sen's (1969) work.

45

We now define Sen's statistic. Let

Vidi+bx) (2.4.14)

denote the value of V^ (2.4.10) based on Y. + bx , Y. +
bx2, ..., Y^^ + bx^. Define

V = (V^ + v^)/2. (2.4.15)

Assuming H^ : &^_^ = o, let 6* denote the Hodges-Lehmann
estimate of the common slope of the two lines based on V.
Define

Vi = V.(Y.-3*x), (2,4.16)

i = 1, 2. Then Sen proposed the statistic

. _ 2 .2

^ ~ -l.^i (2.4.17)

to test Hq: B^__2 = against H^ : &^_^ ^ 0.

The statistic L is a quadratic form in the V., i = l, 2. We

now give an intuitive motivation for the form of L.

The statistic V^ is the value of V. based on the
observations

^il - ^^-^^1' ^12 - ^*-2 ^iN - e*-N-

46

Under H^ : &^_^ = , the 2 lines are parallel and g* is an
appropriate estimator of the corrjr.on slope. In that case,
the transformed observations behave essentially as random
errors fluctuating about zero. Then L, being the sum of
squared random errors with mean zero, has an asymptotic
central chi-squared distribution, and a test of H may be
based on L using this asymptotic null distribution. Under
H^: 6^_2 7^ 0, the estimate B* is not appropriate since the
two slopes are not the same. The transformed observations
will not, in general, have mean zero. Hence the value of L
will be larger than expected under the null, and the use of
the null, central chi-squared critical values will tend to
lead to rejection of H .

PARE Specifics: Alternatives, Regression Co nstants ,
Sequence of Sample Sizes '

In computing the PAREs, we assume a sequence of
alternatives to the null hypothesis H : g = o specified
^y ^N" ^1-2 " ^1-2^^-^ " w/N'a^ as in Theorem 2.2.2.
However, following Hollander (1970) , we consider only
equally spaced common regression constants resulting from
setting c^ = c^ = c, L^ = L^ = L in (2.4.2), with n
observations per line at each of the 2k x-values. Clearly
then, N = 2kn, a^ = c^ (4k^-l) /12 , and

H^: B^_2(N) = 6^_2(2kn) = [o)/ (2kn) ^] [2 (3^) /c (4k^-l) ^] .

(2.4.18)

47

The PARES of the test procedures will be derived under two
different schemes for allowing the sample size N = 2kn to
tend to infinity:

Ca.se 1; Let C be a positive finite constant and let c = c
such that kc, ^ C as k ^ ». Consider n fixed, ^

Case_2: Let c be constant. Consider k fixed, n -> «> ,
resulting in a PARE in terms of k. Then let
k ■> <» in this expression.

Case 1 essentially allows the number of distinct
regression constants (2k) to tend to infinity over a fixed
region of experimentation. Case 2 considers the number of
distinct regression constants fixed while the number of
replicates (n) tends to infinity. The efficiency
expressions derived under case 1 and case 2 will be seen to
be identical. In view of (2.4.18), the rate at which the
sequence of alternatives converges to the null is inversely
proportional to the square root of the sample size, N = 2kn,
for both case 1 and case 2 .

Before proceeding to derive the various PARE's, we
state a theorem due to Noether (1955) as presented by
Randies and Wolfe (1979, p. 147).

Theorem 2.4.1 (Noether ' s Theor em) . Let {S ,.,} and

~ — • n (x)

{T^,(i)} be two sequences of tests, with associated
sequences of numbers { Pg (n (i) ) ^^^ > ' ^ '^T (n' (i) ) ^ ' ^ ^ '

48

2 2

^^S(n(i)) ^^^ ^' ^^^ ^"T(n(i)) ^®^ ^' ^^^ satisfying the

following Assumptions A1-A6:

M. 'n(i) - ^S(n(i))(Qi) .^^ ^n-(i) " ^T(nWi))<^)
''S(n(i))(^^ ^T{n'(i))^'i^

have the same continuous limiting (i ^ ») distribution with
cdf H(.) and interval support when 6. is the true value of

•it:

A2. Same assumption as in Al but with 0. replaced by 6
throughout.

1^00 (J ... (ft \ -i -i.m ^ /n \,— 1

^S(n(i))(Qo) i-^- ^T{n'(i))<%^

A4. ^

de f'^S(n)(^^] = ^s(n) <^) ^"^

fe f^T(n')^^^^ = ^T(n') ^^^

are assumed to exist and be continuous in some closed
interval about 9 = e^, with V^^^^ie^) and y'(^.)(eQ) both
nonzero.

A5.

li^ - S(n(i))'-i^ ^ .^ ^ T(n'(i))'"i^

S(n{i)) ^^0' "^"' ^ T{n'(i)) ^'O'

^6 lim ^'s(n) ^Qq)

^'^' n-°° 2 T = K and

K(„)'^o"'

49

lim ^'T(n') ^^o^

■ , , 2 J- ~ ^T '

where K^ and K^ are positive constants, called the
efficacies of the tests based on S^ and T^, respectively.
Then the PARE of S relative to T is

PARE (S,T) .= -\ . (2.4.19)

^T

Proof ;

See Randies and Wolfe (1979, p. 147).

Note that assuming the equally spaced regression

constants described just prior to (2.4.18) it follows that

conditions I and II hold under either case 1 or case 2.

Hence we need only explicitly assume conditions III and IV

to apply Theorem 2.2.2 which establishes the asympototic

distribution of the Sievers-Scholz statistic under H . We

N
now present the PAREs,

PARE (Sievers-Scholz, Hollander )

Result 2.4.2; Assume the sequence of alternatives {H }
(2.4.18) and the equally spaced regression constants
described just prior to (2.4.18). Also assume conditions
III, IV, and I(h)<". Then the PARE of the Sievers-Scholz
statistic T(0) with respect to Hollander's W under case 1 or
case 2 is

50

2
PARE(Sievers-Schol2, Hollander) = ^[ '^ ^ ^^^ ] . (2.4.20)

■^ 21^ (h)

Verification of Result 2.4.2 ; We apply Noether's

theorem. Assumptions A1-A6 must hold for T(0) and W. For

2

T(0)/a , Theorem 2.2.2 establishes Al and A2 with the

standard normal limiting distribution and standardizing
constants (suppressing the various subscripts and taking

e=6^_2)

y (9) = 4knl (g) 6 , and
a^{8) = 8kn/[c^ (4k^-l)] .

Then A3 , A4 , and A5 follow immediately from the form of
these standardizing constants. Assumption A6 holds and

efficacy (T(0)) = lim y ' (0) / [ (2kn) '^o (0) ]

Itlig) case 1,

(2.4.21)
c(4k^-l)^I(g) case 2.

We note that W is the Wilcoxon signed rank statistic
computed using nk independent random variables with the
distribution of

^l"^2"^3^^4
'1-2 ^ k^

51

As in Randies and Wolfe. (1979, p. 165-166), the assxunptions

A1-A6 can be validated using the equivalent statistic

nk
W/ (^ ■") , in which case

y(6) = (2/(nk-l)) [1-H (kc (-6) ) ]

+ 1 - /H(kc(-t-2e))dH(kct) ,
o^(e) = l/(3kn) ,

the limiting distribution in Al and A2 is the standard
normal , and

efficacy (W) = lim y ' (0) / [ {2kn) ^a (0) ]

N->-<»

6 ^CI (h) case 1,

J,
6 ^kcl (h) case 2 .

(2.4.22)

Then (2.4.20) follows immediately from (2.4.19) of Noether's
theorem.

The first four rows of Table 1 show the value of the
PARE (2.4.20) when the error distribution F is uniform,
normal, double exponential, and Cauchy. The distributions
are listed in order of the increasing heaviness of their
tails, using the measure

F-^.95) - F-^.50) ^ ^.^^_23)

F ^ (.75) - F~^ (.50)

52

Table 1. PARE (Sievers-Scholz , Hollander) and Heaviness of
Tails for Selected Error Distributions.

Distribution

Uniform

Normal

Double Exponential

Cauchy

PARE

1.29
1.33
1.48
2.67

Heaviness of Tails

1.80
2.44
3.32
6.31

CN(e,a^,5,.9)

CN(e,a^,5,.8)

CN(e,a^,5,.7)

CN(e,a^,5,.6)

CN(e,a^,5,.5)

CN{e,a^,10,.9)

CN(e,a^,50,.9)

CN{e,a^,100,.9)

CN(e,a^,500,.9)

CN(e,c^,1000,.9)

1.71
1.94
1.98
1.86
1.70
2.13
2.83
2.96
3.07
3.08

2.82
4.10
5.16
5.34
5.02
3.07
3.68
3.92
4.51
4.78

1) Heaviness of Tails =

F"^(.95) - f"^(.50)
f"^(.75) - f"^(.50)

2) CN(e,a ,ic,t) represents the mixture of two independent

normals. The first, chosen with probability x, has mean

2
6 and variance a'. The second, chosen with probability

1-T, has mean

2 2
and variance k a

53

defined by Crow and Siddiqui (1967), where F~-'-[F(t)] = t.
It is clear that for these four distributions, the PARE of
the Sievers-Scholz statistic with respect to Hollander's
statistic increases with increasing heaviness of tails. To
see whether this behavior persists for other distributions,
we examined the PARE (2.4.20) for a variety of contaminated
normal distributions defined as follows.

Let Z = X with probability t and Z = Y with probability
1-T, where X is N(e,a'") and Y is independently N(e,K^a^).
Then we say Z is distributed as a scale contaminated com-
pound normal which we designate by CN (6 , a^ , k , t) . VJhen F is

2
the CN(e,a ,k,t) distribution, straightforward computations

yield the following formulas for 1(g) and 1(h):

,4, 4-i

,4> 4-1, T ,1

(.)t (1-t)

Kg) = -T^— J {-^^- Y— r^ ' (2.4.24)

TT^a ,tr, [2(i(K^-l)+4)] '

, g (J)tS-Ni-t)^

1(h) = -^^^— ; {-^ J-} . (2.4.25)

^'a /I [2(i(K^-l)+8)] '

In addition to the four common error distributions.
Table 1 shows the value of the PARE (2.4.20) for several
contaminated normal error distributions along with the
heaviness of the tails of these distributions. The formiulas
(2.4.24) and (2.4.25) were used to compute these PAREs while
iterative techniques provided the heaviness of tails of the

54

various contaminated normal error distributions. Figure 1
shows a plot of the PARE (2.4.20) against the heaviness of
tails for all the error distributions in Table 1. The main
purpose of Table 1 and Figure 1 is to show the PARE (2.4.20)
of the Sievers-Scholz statistic T(0) to Hollander's W is
greater than one over a wide range of underlying distri-
butions. Secondarily, we notice in Figure 1 that although
no exact relationship exists, distributions with heavier
tails tend to show higher PAREs.

Under case 2, using the efficacies in (2.4.21) and
(2.4.22), it follows that the PARE of the Sievers-Scholz
statistic to Hollander's statistic before allowing k to tend
to infinity is

4k^-l r I^(q) i

2 '■ 2 ^ '
3k 21 (h)

Thus we see that the PARE is increasing in k, with a minimum

2 2 . .

value of I (g) /2I (h) for k=l . This is likely due to the

fact that Hollander's procedure uses a decreasing proportion

of the available slope estimates as k ->• «>. We have chosen

to present our results after allowing k ->- °° for ease in

interpretation and to avoid the need for a separate

discussion of case 1 and case 2.

PARE (Sievers-Scholz , Rao-Gore)

Result 2.4.3 : Assume the sequence of alternatives {H }
(2.4.18) and the equally spaced regression constants

Figure 1. Plot of PARE (Sievers-Scholz , Hollander) versus
heaviness of tails for selected error
distributions.

1) Heaviness of Tails =

F -^(.95) - f"^(.50)
F~^(.75) - f"^(.50)

2) U = Uniform
N = Normal

E = Double Exponential
C = Cauchy

2

3) CN(e,a ,k,t) represents the mixture of 2 independent

normals. The first, chosen with probability t, has mean
e and variance a . The second, chosen with probability
1-T, has mean 6 and variance k^o^.

T^ = CN(9,a^,5,.9)
T^ = CN(0,a^,5,.8)
T3 = CN(e,a^,5,.7)
T^ = CN(e,a^,5,.6)
T^ = CN(e,a^,5,.5)

K^ = CN(e,a ,10,.9)
^2 = CN(e,a^,50,.9)
K3 = CN(e,a^,100,.9)
K^ = CN(e,a^,500,.9)
<5 = CN(6,a^,1000,.9)

PARE I
J. 5 +.

56

2.5 ■»•

Kit — K5

T4

1.5 +

0.5 +

TS

Heaviness of TpIIs

-'^F'»wi»w^w^gWB»»»W'^Ba8waic«iMBaw »MML i ai « ». i T ii|ii tu-uj- i M i[i^ j» l>i iaij i j ^'MW[;^B^

57

described just prior to (2.4.18). Also assume conditions

III and IV. Then the PARE of the Sievers-Scholz statistic
T(0) with respect to the Rao-Gore statistic U under case 1
or case 2 is

PARE (Sievers-Scholz, Rao-Gore) = 4/3. (2.4.26)

Verification of Result 2.4.3 ; As in Randies and Wolfe
(1979, p. 170-171), the assumptions A1-A6 of Noether's
theorem can be validated for the statistic U + (nk (nk+1) /2) ,
equivalent to U, in which case

y(e) = (nk)^/G(kc(t+e))dG(kct) + (nk (nk+1) /2) ,
a^(e) = n^k^/6,

the limiting distribution in Al and A2 is the standard
normal , and

efficacy (U) = lim n ' (0) / [ (2kn) ^a (0) ]

3 '01(g) case 1,

(2.4.27)

3 ''kcl (g) case 2.

We showed A1-A6 hold for T(0) in the previous verification
and gave the efficacies of T(0) in (2.4.21). Thus (2.4.26)
follows imiD.ediately from (2.4.19) of Noether's theorem.

®

58

PARE(Sievers-Scholz, Classical Least Squares)

Result 2.4.4 : Consider the classical least squares
theory t-test, as specified by Hollander (1970) . Assume the
sequence of alternatives {E^} (2.4.18) and the equally
spaced regression constants described just prior to

(2.4.18). Also assume conditions III, IV, and o

2
a = Var(E^)<«>. Then the PARE of the Sievers-Scholz

statistic T(0) with respect to the classical t-test

statistic under case 1 or case 2 is

PARE (Sievers-Scholz, classical) = 24a^[I^(g)]. (2.4.28)

Verificat ion of Result 2.4.4 ; Let B^ and ^2 denote the

least squares estimates of 6^ and B^' respectively, and let

2
s be the residual mean square error (see (3.1) of Hollander

(1970, p. 389)). Then the form of the t-test statistic used

to test Hq: Bj^_2 = under the specified equally spaced

regression constants is

e - 6

^ - 1 2

t - 2 • (2.4.29)

r 12s ,h

2 2-'
nkc (4k^-l)

Using the fact that s is a consistent estimate of a^ under
B^^ and a discussion similar to that in Randies and Wolfe
(1979, p. 164-165), it follows that assumptions A1-A6 of
Noether's theorem hold for the statistic t with

iJiii II WH»*Wiui I " wii ■ ■ "I, j-ionwif ij .11 iiiH»

59

y{e) = ^- ' ^"'^ (2.4.29)

r 12a^ .h

^ 2 ? ^

nkc (4k -1)

a^(e) - 1.

The limiting distribution in Al and A2 is the standard
normal and

efficacy (t) = lim y ' (0) / [ (2kn) 'a (0) ]

[ — 5-] case 1,

6a

, c^(4k^-l) ,^

[ ^ 5—^] case 2.

24a^

(2.4.30)

A1-A6 hold for T(0) as shown in the verification of Result
2.4.2 with the efficacies given by (2.4.21). Thus (2.4.28)
follows immediately from (2.4.19) of Noether's theorem.

The efficiency (2.4.28) is identical to the familiar
PARE of the two-sample Wilcoxon-Mann-Whitney test with
respect to the two-sam.ple normal theory t-test when the cdf
of the underlying error distribution is G. Hence, the
PARE (Sievers-Scholz, classical) > 0.864 for all G (see
Hollander (1970) for a proof that in this case the inequal-
ity is strict). Also, this PARE equals 0.955 when G is
normal and is greater than one for many non-normal G.

60

PARE (Sievers-Scholz.Sen)

Result 2.4.5: Assume the sequence of alternatives {H,^}
(2.4.18) and the equally spaced regression constants
described just prior to (2.4.18). Also assum.e the following:

1) Conditions III and IV.

2) I* (F) < ~ as in (2.4.11), where F is the under-
lying error distribution.

3) The score function ({) (u) is an absolutely continuous
and nondecreasin.g function of u, < u < 1, that is
square integrable over (0,1).

Let pCy,!})) be defined as in (2.4.13). Then the PARE of the
Sievers-Scholz statistic T(0) with respect to Sen's (1969)
statistic L under case 1 or 2 is

PARE (Sievers-Scholz, Sen) = J"^ -*• ^^^ ' (2.4.31)

p^ (¥,<!)) I* (F)

Verification of Result 2.4.5 :

Sen's statistic L has an asymptotic (N->-<») chi-squared

distribution under H (Sen, 1969, p. 1676). Using results

_*
from Sen's work we define a statistic, V , that has an

asymptotic normal distribution under H and whose square

(multiplied by a constant) is asymptotically equivalent to L

_*

under H . Since the square of V (multiplied by a constant)

has the same asymptotic distribution under H as L, we use

_*

V in applying Noether's theorem to derive efficacy

?^s^sa>B9!gi^~ap;> i i -i ji gi pt* ' »n, <, !■ hjiuhii

61

expressions for Sen's test and the PARE (Sievers-Scholz ,
Sen) .

Let B^ denote the Hodges-Lehmann estimate of 3, based
on V^ (2.4,10), i=l,2, and let 6* = 1/2(6* + 6*)* From
equations (3.10) and (3.20) in Sen (1969, p. 1673, 1675) it
follows that

a = N'a (6 -3*) = 0^(1) and
^ J. p

as N^" under H^, i=l,2. Using these definitions of a and b
and the notation (2.4.14) we apply Leimna 3 . 2 in Sen (1969,
p. 1674) , which is given in Chapter Three of this work as
Lemma 3.4.3, yielding

V.(Y.-B X) - V.(Y.-6.x)

= N'a^(6*-3*)p(¥,(|,) [I*(F)]^ + o (1)

as N-^- under H^. Equation (3.22) in Sen (1969, p. 1675)
states

|Vi(Y.-e*x)| = o (1)

•^'"'''Wl f .l ii . ' l"_ l «.>--'_P_» i|«l . l lv « LM i«»

62

and so, applying this to the previous result.

^i " Vi(Ii-^*2S)

= N'a^(S*-6*)p(Y,<(,) [l*(F)]'^ + o (1)

as N->co under H^. Applying equation (3.19) in Sen (1969,
p. 1675) ,

N^a^{6*-3*) = o (1) ,

we have

^i " N'a^(B%B*)p (T,^) [i{F)]^ + o (1)

as N^-<» under H^. It follows that

63

2

L = p^(Y,<f)I*(F)NaJ E (e*-3*)^ + o^(l)

^ i=l ^ P

= p^(^,(j))I*(F)Na^%(3*-32)^ "^ °p^^^

as N->» under H„. We define the statistic
N

-* ■ H. * *

V = (N/2) cr^(3-^-62) (2.4.32)

and note that the asymptotic distribution of

2 * -* 2
p ('!',(()) I (F) (V ) under H is the same as that of L. From

Lemma 3.4 in Sen (1969, p. 1676),

p(^,<i>) [I*(F)]^[V*-(N/2)'^a^(6^-S2)] t N(0,1)

as N-»-« under H . Thus assumptions A1-A6 of Noether's

_*
theorem hold for the statistic V with

y(0) = [c(4k^-l) V(2(3'))] G, and

o^ie) = [p^(^,<^)i^ (F)] ^

64

The limiting distribution in Al and A2 is the standard
normal and

efficacy (V ) = lim y ' (0) / [ (2kn) ^o (0) ]

h, ,.%

[Cp(T,.^) (I (F))^]/6^ case 1,

(2.4.33)

.[c {4k^-l) ^p (¥,<(,) (I* (F) ) '^j / [2 (6^) ]

The assumptions A1-A6 hold for T{0) as shown in the
verification of Result 2.4.2 with the efficacies given by
(2.4.21). Hence (2.4.31) follows immediately from (2.4.19)
of Noether's theorem.

To evaluate the expression (2.4.31), we first note the
following, which results from the definition of A^ in
(2.4.9) and q(1 ,^) in (2.4.13) :

P^('i',<).)I*(F) = [ / ¥(u)(j,(u)du]2/A^

^ 2 1

= [ / >l'(u)(|)(u)du]"^/[ / [ Mu) -<()*] ^du] .

(2.4.34)

Suppose we assume Mu) = u, < u < 1. Scores resulting
from this choice of score function are called Wilcoxon

65

scores. Direct computations show that in this case

1 2

/ [(j)(u) - ^*]^du = 1/12. (2.4.35)

Using a derivation similar to one in Randies and Wolfe
(1979, p. 308) we show that when ^ (u) = u, the numerator of
the right-hand side of (2.4.34) equals I^(f):

/ Y(u)(|) (u)du = - / u[f' (F "-(u) )/f (F ^(u))]du.

(2.4.36)

Let t = F (u) , resulting in

1 Ul)

/ 4'(u)(i. (u)du = - / F(t)f'(t)dt, (2.4.37)
UO)

where F(?(p)) = p. Now let u = F(t), dv = f'(t)dt and apply
integration by parts to (2.4.37):

^ Ml) ^^^) 2

/ ^(u)((.(u)du = - {[F(t)f (t)]^ ; - / f^(t)dt}

^^^> UO)

= [F(UO))f (5(0))] - [F(C(l))f (UD)] + / f^(t)dt.

5(0)

(2.4.38)

If we assiime the support of F is [a,b] where a < b and

f(x) ^-Oasxfborx + a, then we take 5(0) = a, 5(1) = b

in (2.4,38) and

66

1 b

/ Y (u) (}) (u) du = / f^ (t)dt = 1(f) . (2.4.39)

a

As Randies and Wolfe (1979, p. 313) state, this same form
(2.4.39) can be obtained under more general assumptions.

Hence substituting (2.4.35) and (2.4.39) into (2.4.34)
it follows from (2.4.31) that under ^ (u) = u, H , and
assumed regularity conditions,

2
PARE(Sievers-Scholz, Sen) = ^^ ^^^ . (2.4.40)

I (f)

Table 2 gives values of the PARE (2.4.40) for four
common error distributions. We see that when compared to
the Sen statistic using Wilcoxon scores, the Sievers-Scholz
statistic achieves a PARE close to (or equalling) one for
error distributions having light to moderately heavy tails
(uniform, normal, double exponential) . However the
Sievers-Scholz statistic has poor PARE under the Cauchy
distribution which has very heavy tails. That the
asymptotic performance of Sen's test is better than other
tests is not surprising since Sen's test is the rank test
that maximizes the efficiency relative to the likelihood
ratio test. However, Sen's test not only requires iterative
calculations, but is distribution-free only asymptotically.
More discussion of the relative merits of Sen's test and the
Sievers-Scholz testd will be given in Section 2.6.

67

Table 2. PARE (Sievers-Scholz , Sen) for Selected Error
Distributions Assuming (j) (u) = u.

Distribution

Uniform

Normal

Double Exponential

Cauchy

PARE (Sievers-Scholz, Gen)

0.89
1.00
0.78
0.50

Summary of PARE Results

In conclusion, PAREs derived under equally spaced
regression constants favor the Sievers-Scholz approach over
the other two exact, nonparametric competitors due to
Hollander and Rao and Gore. Specifically, the PARE of the
Sievers-Scholz statistic with respect to Hollander's
statistic is greater than one over a wide range of
underlying error distributions. Indeed this efficiency
frequently exceeds two. Even more interesting is the fact
that the PARE of the Sievers-Scholz statistic with respect
to the Rao-Gore statistic is 4/3 for all error distributions
(subject to certain regularity conditions required to derive
the PARE) . The Sievers-Scholz statistic achieves the
familiar PARE (2.4.28) when compared with the classical
least squares theory t-test. Although the PARE of the
Sievers-Scholz statistic with respect to Sen's statistic is

68

less favorable under the heavily tailed Cauchy distribution,

we noted that Sen's test is distribution-free only
asymptotically. V7e discuss these two methods further in
Section 2.6.

Note that under equal spacing, the PARE of the Sievers-
Scholz statistic to the Theil-Sen statistic (2.1.4) dis-
cussed in Section 2.1 is one (Sievers, 1978). Thus all of
the previous ARE comparisons also apply when using the
(zero-one weighted) Theil-Sen statistic in place of the
Sievers-Scholz statistic. The advantage of the Sievers-
Scholz approach over that of Theil and Sen will appear in
the Monte Carlo comparisons under unequal spacing of the
regression constants discussed in Section 2.6.

2.5 Small Sample Inference
Since the test of parallelism and the confidence
intervals for g^^^ presented in Section 2.3 depend on
asymptotic theory, they are generally only applicable with
moderately large samples. In this section we discuss exact,
distribution-free tests of H^ : g^^^ = and corresponding
exact confidence intervals for the slope difference, B
These tests continue our basic approach of applying the
method of Sievers and Scholz to the two line setting, again
assuming comm.on regression constants.

Specifically, we discuss two related techniques. The
first utilizes the exact distribution of the Sievers-Scholz
statistic TiO) with the optimum weights, a = x , under

69

the null hypothesis H^ : 3-,_2 = 0. This null distribution
depends on the chosen regression constants, and hence must
be recalculated for each design. The second technique is a
straightforward application of the Theil-Sen approach to the
two line setting, as discussed in Section 2.1. Since the
required null distribution is essentially that of Kendall's

o

tau, tabled critical values are readily available for small
sample sizes. Thus the second technique has favor under
sample sizes sm.all enough to discourage the use of
asymptotic results, yet too large to allow the computation
of the null distribution of T(0) required by the first
technique.

Let us now discuss in detail the two small-sample
techniques we have proposed. First, recall the basic linear
model (2.1.1) we have assumed, the resulting distribution of
the differences Z. given by (2.1.3), and the representation
of T(0) in terms of the ranks of the Z's given by setting
b = in (2.2.1) :

N
T(0) = (2/N) Z [Rank(Z.)x ] - (N+l)x. (2.5.1)

j = l ^ ^

Suppose that for line i (i=l,2) the underlying errors, E. .,
j=l,...,N, are independent and identically distributed
(i.i.d.). The distribution of the line 1 and line 2 errors

need not be the same, but assume these two distributions are
continuous. Let ^ denote the set of N! permutations of

70

(1,2,...,N). Then under H^ : 3-|^_2 = 0, the Z's are i.i.d.
and hence the vector of ranks of the Z's is uniformly
distributed over ^^ (Randies and Vvolfe, 1979, p. 37).
Consequently, in view of the representation (2.5.1), the
null (B^_2 = 0) distribution of T(0) is uniformly
distributed over the N! values of T(0) obtained by permuting
the ranks of the Z's among themselves. For example, if for
r = (r^,. . . ,r^) e|^,

N
t„ = [ (2/N) E (r.x.)] - (N+l)x ,
- j=l ^ ^

then the distribution of T(0) under H. : g = is given by

Pq {T(0)=t^} = 1/N!, rti^, (2.5.2)

where, as before, ^^i ' } denotes the probability calculated
assiiming B^^^ = te Having tabulated this null distribution,
a test of H- is conducted as follows.

If t^ is a constant determined from (2.5.2) such that
Pq {T(0)>_t^} = a, then the test of H^ : &^_^ = against
Hq: B^_2 > which rejects H„ if

T(0) 1 t^ (2.5.3)

is an exact level a test. Rather than simply state whether
a test has accepted or rejected a null hypothesis at a

i MU i u 'ww LHJ ^ - ' j ^ tig ii j iw ^i^ tt Mw u ' L iii ^T, v ftai ri ^Sf*

71

particular level, one might wish to report the attained
significance level, the lowest significance level at which
the null hypothesis can be rejected with the observed data,
If we observe T(0) = t(0), then the exact test (2.5.3) has
an attained significance level of

p = [ (number of r in P) /N! ] ,

where P = {re^,: t(0)£t^}. Two-tailed tests can be
obtained by the usual modifications.

Lst t^ -2 be a constant defined analogously to t :

^0 ^-^a/2^^(0)^^a/2> = 1 " -

Applying the argximent and notations used to derive (2.3.3),
it is easily seen that

[j;^(%(l-(Nt^/2/^--)))' jZ^(^a+i^t^^^/x..))))

(2.5.4)

is an exact 100 (l-a) percent confidence interval for 6.
In terms of the slopes S^^ = (Z -Z ) /x , 1 j< r < s <_ N,
we can write this interval as [S""^, S^) , where

Sg = min{S^g: J (S^^) >% (1- (Nt^^2/^- • ) ) ' l<r<s<N},

and

MB'.m^.r^Tl'^igi l

72

Sq = min{S^^: J (S^^) >h {1+ (^t^^^/x. .) ) , l.<r<s<N} .

(2.5.5)

Computation of a null distribution based on permuta-
tions of the observations (or ranks) such as the one in
(2.5.2) has become possible under small sample sizes with
the speed of today's computers. At the University of
Florida we found that the necessary calculations were
feasible when there were N=8 or fewer regression constants
per line. In this case the entire null distribution of T(0)
could be tabulated at a cost of about \$2.00 to the user. We
were accessing a system operating an IBM 3081 with MVS/XA
and an IBM 3033 with OS MVS/SP JES2 Release 3. With more
than 8 regression constants per line, the cost of tabulating
the null distribution becomes prohibitive (at least \$30.00
when N=10, for example). However, in dealing v/ith similar
problems, Pagano and Tritchler (1983) and Mehta and Patel
(1983) give algorithms that greatly reduce the amount of
computation involved. Although their results do not apply
directly to this problem, we anticipate the exact test
(2.5.3) and computation of the confidence interval (2.5.4)
will soon become feasible for larger sample sizes due to the
development of similar efficient algorithms and the steadily
increasing speed of computer hardware.

At the present time we could resort to estimation of
the null distribution (2.5.2) of T(0) based on a random
sample of permutations of the observed rank vector. Boyett

■ "" i^u f^ gJC .1 1 ii a^ i qm BBj rL P'!WqiMWi^-*?j g, 'g"' . » ajt'j » i ■■ wn nn mngi w em w * t atm m nuiim

73

and Shuster (1977) and Ireson (1983) report good approxima-
tions resulting from the use of such a sampling of permu-
tations to approximate a permutation distribution. However,
there is a possibility that approximate procedures such as
these suffer a loss of power due to the restricted sample
space (Dwass, 1957) . Also, there is the problem of specifi-
cation of the number of permutations that must be sampled to
achieve adequate approximations of the null distribution.
Clearly, more study is needed before this technique can be
recommended without reservation.

Another method of overcoming the computational problem
associated with the exact test under larger sample sizes is
to replace the Sievers-Scholz statistic by the Theil-Sen
statistic based on the differences Z . , as described in
Section 2.1, Recall that this consists of using the weights
^rs " sgn(x^^) in the expression for T (b) given by (2.1.2).
The null distribution is essentially that of Kendall's tau,
which has been tabulated for many values of N. These tabled
critical values and a precise specification of the Theil-Sen
test and confidence interval are given in Hollander and
Wolfe (1973) . This second approach is appropriate when an
exact, distribution-free technique is desired but the number
of regression constants per line is too large (N>8, at our
facility) to allow complete enumeration of the null distri-
bution (2.5.1) of T(0). If the regression constants are
highly unequally spaced, the method based on Sievers-Scholz
procedure probably has greater power, but the results of a

74

simulation study discussed in the next section indicate that
in many situations the difference in power of the two
techniques is slight.

One detail that has not been mentioned in this section
is how to deal appropriately with ties in the data. If ties
occur among the Z's when applying the first technique,
simply use all permutations of the vector of midranks in the
computation of the null distribution of T(0). The null
distribution computed is the conditional null distribution
of T(0), given the observed midranks, and the exact, dis-
tribution-free properties of the test and interval are
retained. We assume the Theil-Sen approach is only being
applied when the sample size prohibits computation of the
exact null distribution of T(0). In this case, modifica-
tions of the Theil-Sen approach based on Kendall's tau in
the presence of ties are referenced by Hollander and Wolfe
(1973, p. 192), but they do not retain the exact nature of
the test and confidence interval.

As discussed in Section 2.4, the Hollander and Rao-Gore
methods also provide exact, distribution-free tests of
HqJ 3^_2 = and confidence intervals for the slope differ-
ence, 3-j^_2. The null distributions required to use their
methods have been tabulated and are readily available for
several sample sizes. Hence these methods are competitors
to the exact, small sample techniques proposed in this
section. Monte Carlo comparisons of the power of these
tests are given in the next section.

75

2.6 Monte Carlo Results
To compare the powers of the test statistics discussed
in the previous sections, we conducted a Monte Carlo study.
The study concentrated on unequally spaced regression
constants for the following reasons. The asymptotic rela-
tive efficiency results in Section 2.3 indicate superiority
of the Sievers-Scholz procedure in a wide variety of cases
when the regression constants are equally spaced. Since the
structure of the Sievers-Scholz statistic T (b) (2.1.5)
utilizes information about the spacing of the regression
constants, one would expect that its relative performance
improves when unequal spacing is used. Hence, we were
particularly interested in comparisons of the test statis-
tics under unequal spacing of the regression constants. We
begin our discussion with a description of the sample sizes,
regression constants, error distributions, and parameter
values used in our simulation study.

Choice of Regression Constants

Recall from Section 2.4 that Hollander's technique
requires a scheme for pairing the regression constants on
each line to form slope estimates. Although Hollander
clearly specifies the pairing scheme (2.4.3) under equal
spacing (2.4.2) of the regression constants, there exists
some ambiguity in the choice of such a scheme when the
regression constants are unequally spaced. To avoid this

76

ambiguity, we use what we call mirrored spacing, which we
now describe.

Consider a smooth nondecreasing function t(-) defined
over the interval [0,1/2]. Suppose t{.) maps this interval
onto itself with t(0) =0 and tCl/2) = 1/2. Then we define
2k regression constants, x^, x^, ..., x^j^, over the interval
[0,1] by the relations

"m = ^(k?l) '

^m+k = ^^m + 1/2 '

for m = 1, ..., k. Multiplication by a scale factor can be
used to make these constants fall over a more natural range.
We use the term mirrored spacing since, by definition,

'm+k ~ ■^m "^ "'■/^ for ^ = 1/ , k, and so the arrangement of

the set of x's {x^^^^, ..., x^^^} over [%,!] is identical to
the arrangement of the set {x^, ..., x^^} over [0,^]. when

constructing Hollander's statistic we pair responses at x

m

with those at x^^^, thus conforming to Hollander's recom-
mendation to choose a pairing scheme approximating that used
with equal spacing. Mirrored spacing also allows the use of
the Rao-Gore statistic, which is not generally applicable
with unequally spaced regression constants.

We experimented with t(.) of the form t (u) = au^, where
a is chosen to satisfy t(l/2) = 1/2 and i is a positive
integer. Note that i = 1 results in two groups of equally

X

77

spaced constants over the intervals (0,1/2) and (1/2,1). As
i increases, the regression constants tend to group closely
just below 1/2 and 1. For our simulation study we selected
1-3, t(u) = (1/3) 'u . This choice of i results in regres-
sion constants that depart sufficiently from equal spacing
without the excessive clumping observed under larger values
of i.

Selection of Err or Distributions and Parameter Values

To generate simulated random variates we used the
Fortran subroutines of the International Mathematical and
Statistical Library (IMSL) . We selected four error dis-
tributions: uniform, normal, double exponential, and
Cauchy. In terms of the heaviness of tails, these dis-
tributions cover a broad range. They are listed above in
order of increasing heaviness of tails, from the uniform
distribution which has very light tails to the Cauchy
distribution which has very heavy tails. The standard
normal distribution (N(0,1)) was used, and scale factors of
the other distributions were selected such that the proba-
bility between -1 and 1 was the same for all four distribu-
tions .

The values of &^_^ at which the power was estimated
were determined by selecting multiples of the estimated
standard deviation of the difference of the least squares
estimates of g^ and e^ in each case. These multiples were
chosen to achieve a wide range of power.

78

Designs and Computational Details

There are basically two parts to the Monte Carlo study
presented here. In the first part we used moderately large
samples and applied approximate tests based on asymptotic
theory. The second part of the simulation used small sample
sizes and exact nonparametric tests. The form of the
classical t-test was the same in both parts of the Monte
Carlo simulation.

For the first part of the Monte Carlo study, dealing
with moderately large samples, we chose two designs which we
call design A and design B. In presenting the sample sizes
used in each design, we give the number of regression
■constants per line. Since we are assuming two lines, the
total sample size is twice of what we give below. Design A
consists of 3 replicates at each of 20 distinct regression
constants, resulting in 60 observations per line. The
20 distinct regression constants were selected by using
mirrored spacing as described and multiplying by a scale
factor of 20. The resulting regression constants are listed
in Table 3.

Design B uses 30 distinct regression constants, with
one observation per line taken at each of these constants.
The constants for design B were selected by generating
30 random numbers between zero and one, and multiplying them
by a scale factor of 30. We wanted one design chosen to
allow power comparisons of the tests under regression
constants whose spacing followed no structured pattern, and

79

Table 3. Regression Constants Used for Designs A and

B,

Design A

Design B

X.

X,

X,

X-

^10

X,

X,

X,

X,.

X,

X.

Xr

X,

X

X

X

'10

11

12
'l3

14
15

= 0.0075
= 0.0601
=. 0.2029
= 0.4808
= 0.9391
= 1.6228
= 2.5770
= 3.8467
= 5.4771
= 7.5131

= 0.65
= 1.16
= 5.46
= 6.09
= 8.34
= 8.52
= 9.50
= 10.83
- 13.61
= 13.84
= 13.93
= 15.20
= 15.42
= 15.58
= 16.51

X

X

X

X

X

X

11

'12
13
14
15
16
17
18
19

X

20

X

X

X

X

X

X

X

16
17
18
19
20
21
22
23

X

X

X

'24
'25
26
27
28
'29
'30

= 10.0075
= 10.0601
= 10.2029
= 10.4808
= 10.9391
= 11.6228
= 12.5770
= 13.8467
= 15.4771
= 17.5131

= 17.59
= 18.81
= 18.83
= 19.53
= 19.99
= 20.13
= 20.78
= 21.01
= 22.76
= 23.78
= 25.86
= 26.37
= 26.59
= 28.06
= 28.92

1) For Design A, 3 responses per line were observed at
each of the 20 regression constants.

2) For Design B, 1 response per line was observed at each
of the 30 regression constants.

80

hence this accounts for our departure from mirrored spacing
in this case. The regression constants for design B are
listed in Table 3. To compute Hollander's statistic under
design B, we used the usual grouping scheme, pairing the
response at x^ with that at ^^+^5' ^^^^^ m = , 1, ..., 15.
Note that given the arbitrary spacing of the regression
constants, the Rao-Gore statistic is not applicable for
design B.

All tests applied in the simulations under design A and
B used a nominal level of a = .05, and in each case the null
hypothesis H^ : ^-^-2 ~ ° ^^^ being tested against the one-
sided alternative H^ : ^^_^ > 0. The tests used were the
Sievers-Scholz test described in Section 2.3, the Theil-Sen
test (based on T* (b) in (2.1.2) with a = sgn (x -x ) ) , the
Hollander test based on the statistic W defined in
Section 2.4, the Rao-Gore test (design A) based on U defined
in Section 2.4, and the t-test based on the difference of
the least square estimates of 6, and g-- Each of these
tests were employed at an approximate a = .05 level
utilizing their respective asymptotic distributions.

For comparisons of the various procedures under small
samples we selected three designs, which we call design C,
design D, and design E. All three designs consist of one
response per line at each of the regression constants.
Sample sizes per line of 6 , 8, and 12 were used in
designs C, D, and E, respectively. Mirrored spacing was

81

used for all three designs. The design points were multi-
plied by scale factors of 10, 10, and 15 for designs C, D,
and E, respectively. The resulting regression constants for
these three designs are given in Table 4 .

When using designs C and D, the exact tests of
^0" ^1-2 ^ ° against H^ : &^_^ > Q associated with each of
the four nonparametric procedures were used. The exact
Sievers-Scholz and Theil-Sen tests as discussed in Sec-
tion 2.5 were applied. Thus the exact distribution of the
Sievers-Scholz statistic T{0) was first computed for designs C
and D to determine appropriate critical values.
An example of a portion of this distribution for design C is
given in Table 5. Exact versions of the Hollander and
Rao-Gore tests rely on the exact null distributions of the
Wilcoxon signed rank and Wilcoxon-Mann-Whitney statistics,
respectively, as tabulated and discussed in Hollander and
Wolfe (1973) . Randomization was used to bring these exact
procedures to the same level. The natural a-levels of the
tests were compared to select a level for each design at
which the amount of randomization needed was minimal. A
nominal level of a = .125 was used for design C while
a = .057 was selected for design D.

With 12 observations per line, design E does not allow
the feasibility of the exact Sievers-Scholz test. Instead,
we replaced this with an approximate procedure, randomly
selecting 10,000 permutations of the rank vector and com-
puting the proportion of these perm.utations resulting in a

"-iM-^ti^*;, ■ |HMiH|ni(^i9i«

82

Table 4. Regression Constants Used for Designs C, D,

and E.

Design C ^i = 0.078 x^ = 5.078

x^ = 5.625

Xg = 7.109

Design D x^ = 0.040 x^ = 5.040

Xg = 5.320

x^ = 6.08

Xg = 7.560

Design E x^ = 0.022 x^ = 7.522

Xg = 7.675
Xg = 8.090
x^Q = 8.899
x^^ = 10.233
x^2 = 12.223

^1

=

0.078

^?

=

0.625

^3

=

2.109

^]

=

0.040

^2

=

0.320

^3

=

1.080

^4

=

2.560

^1

=

.022

^7

=

.175

^3

=

.590

^4

= 1

.399

^5

= 2

.733

^6

= 4

.723

For Design C, D, and E, one response per line was observed
at each design point.

^*=«!*«¥Va3»¥^-^-^i|||)l .

83 *

Table 5 . Upper Portion of the Exact Null Distribution of
T(0) when using Design C.

t(0)

Pq {T{0) > t(0)}

6.64062
6.69269
6.82289
6.82292

6.87499
7.00519
7.13542
7.18749
7.31769
7.36979
7.49999
7.68229
7.81249
7.86459
7.99479
8.17709

0.0583
0.0542
0.0528
0.0514
0.0486
0.0472
0.0444
0.0431
0.0389
0.0347
0.0333
0.0236
0.0194
0.0167
0.0139
0.0111

-^i-t*ut,rwi»(i«w«<**»rif--»"!**'*^'F— -

84

value of T(0) larger than the one calculated from the
observed data. Rejection of H^ : Q^_^ = o in favor of
^1' ^1-2 ^ ° occurred when this proportion was less than or
equal to the nominal level, which was set at a = .047.
Recall that this type of approximate procedure was discussed
in Section 2.5. The implementation of this approximate
procedure was facilitated by an algorithm due to Knuth
(1973) , which presents a one-to-one association between the
integers 1,...,N! and the N! permutations of (1,...,N).
Because of the cost of this approximate Sievers-Scholz
procedure, the number of simulations for design E was set at
500, and only two error distributions were selected (normal
and Cauchy) .

Discussion of Results

Empirical levels and powers of the Sievers-Scholz,
Theil-Sen, Hollander, Rao-Gore, and classical tests under
design A are presented in Tables 6, 7, 8, and 9, for the
uniform, normal, double exponential, and Cauchy distribu-
tions, respectively. Empirical levels vary but generally
remain within two standard errors of the nominal .05 level.
It is seen that in terms of their power, the Sievers-Scholz
and Theil-Sen tests uniformly dominate the Rao-Gore and
Hollander tests. The power of the classical t-test is
highest under uniform and normal errors, but falls below the
powers of Sievers-Scholz and the Theil-Sen tests for double
exponential errors. The t-test has the lowest power of the

85

Table 6. Empirical Power Tiir.es 1000 Under Design A for the
Uniform Distribution (a=.05).

h-2

.024

.064

.096

Sievers-Scholz

058

192

675

934

Theil-Sen

059

2

- 18

14

Hollander

063

4

- 57

- 35

Rao-Gore

061

- 15

- 58

- 36

Classical

061

17

71

31

The first row and first column give the empirical power
times 1000. Entries in the rem.ainder of the table are
expressed as the difference from the corresponding Sievers-
Scholz result, with a negative value indicating a lower
power than Sievers-Scholz.

Table 7. Empirical Power Times 1000 Under Design A for the
Normal Distribution {a=.05) .

Sievers-Scholz 041 217 561 861

Theil-Sen 036 5 - 21 - 15

Hollander 046 - 10 - 70 - 51

Rao-Gore 041 - 25 - 41 - 53

Classical 042 33 27 26

The first row and first column give the empirical power
times 1000. Entries in the remainder of the table are
expressed as the difference from the corresponding Sievers-
Scholz result, with a negative value indicatira a lower
power than Sievers-Scholz.

86

Table 8. Empirical Power Times 1000 Under Design A for the
Double Exponential Distribution (a=.05) .

^1-2

.032

.064

.096

Sievers-Scholz

052

234

476

808

Theil-Sen

043

- 16

6

- 14

Hollander

057

- 46

- 69

- 116

Rao-Gore

055

- 29

- 53

- 67

Classical

049

- 16

- 32

- 22

The first row and first column give the empirical power
times 1000. Entries in the remainder of the table are
expressed as the difference from the corresponding Sievers-
Scholz result, with a negative value indicating a lower
power than Sievers-Scholz.

Table 9. Empirical Power Times 1000 Under Design A for the
Cauchy Distribution (a=.05) .

^1-2

.064

.096

.160

Sievers-Scholz

052

394

590

896

Theil-Sen

052

- 13

1

1

Hollander

053

- 140

- 245

- 275

Rao-Gore

062

- 41

- 44

- 31

Classical

043

- 289

- 451

- 667

The first row and first column give the empirical power
times 1000. Entries in the remainder of the table are
expressed as the difference from the corresponding Sievers-
Scholz result, with a negative value indicating a lower
power than Sievers-Scholz.

87

five tests under the heavily tailed Cauchy distribution.
The Sievers-Schclz test tends to have higher power than the
Theil-Sen test, but the difference is not pronounced.
Comparing Hollander's test with that of Rao and Gore, we see
that the power of the Rao-Gore test dominates the power of
Hollander's test for the two heavily tailed distributions
(double exponential and Cauchy) but they perform about
equally under normal and uniform errors. A summary of all
the Monte Carlo results will be given at the conclusion of
this section.

Tables 10, 11, 12, and 13 present empirical levels and
powers of the Sievers-Scholz , Theil-Sen, Hollander, and
classical tests under design B for the uniform, normal,
double exponential, and Cauchy distributions, respectively.
Empirical levels of the tests appear somewhat depressed, but
generally fall within two standard errors of the nominal
.05 level. For the uniform, normal, and double exponential
distributions, the order of the tests in decreasing power is
classical, Sievers-Scholz, Theil-Sen, and Hollander. The
dominance of the classical approach lessens with increasing
heaviness of tails of the ■ error distribution. For Cauchy
errors the classical test again falls into last place, with
the Sievers-Scholz and the Theil-Sen tests exhibiting
highest powers.

For design C, Tables 14, 15, 16, and 17 present empiri-
cal levels and powers under uniform, normal, double
exponential, and Cauchy distributions, respectively.

Table 10

88

Empirical Power Times 1000 Under Design B for the
Uniform Distribution (a=.05).

^1-2

.034

.068

.102

Sievers-Scholz
Theil-Sen
Hollander
Classical

039
036
037
043

253

- 16

- 62

37

651

- 18

- 161

76

934

- 12

- 89
31

The first row and first column give the empirical power
times 1000. Entries in the remainder of the table are
expressed as the difference from the corresponding Sievers-
Scholz result, with a negative value indicating a lower
power than Sievers-Scholz.

Table 11. Empirical Power Times 1000 Under Design B for the
Normal Distribution (a=.05).

Sievers-Scholz
Theil-Sen
Hollander
Classical

042
038
055
046

209
20
24
25

569

- 28

- 128

48

878
18

154
26

The first row and first column give the empirical power
times 1000. Entries in the remainder of the table are
expressed as the difference from the corresponding Sievers-
Scholz result, with a negative value indicating a lower
power than Sievers-Scholz.

^•"•^^rv* •■•«J'f ■"** c --'I'-* ■»— ^ J-

89

Table 12. Empirical Power Times 1000 Under Design B for the
Double Exponential Distribution (a=.05).

Sievers-Scholz
Theil-Sen
Hollander
Classical

047
039
045
053

201

4

- 21

18

495

- 24

- 137

2

753

- 23

- 171

The first row and first column give the empirical power
times 1000. Entries in the remainder of the table are
expressed as the difference from the corresponding Sievers-
Scholz result, v/ith a negative value indicating a lower
power than Sievers-Scholz.

Table 13. Empirical Power Times 1000 Under Design B for the
Cauchy Distribution (a=.05).

Sievers-Scholz

044

225

577

770

Theil-Sen

047

6

5

7

Hollander

041

- 90

- 250

- 324

Classical

043

- 133

- 377

- 455

The first row and first column give the empirical power
times 1000. Entries in the remainder of the table are
expressed as the difference from the corresponding Sievers-
Scholz result, with a negative value indicating a lower
power than Sievers-Scholz.

•mtfimmmmmspm*'^

90

Table 14. Empirical Power Times 1000 Under Design C for the
Uniform Distribution {a=.125).

Sievers-Scholz 121 424 824 977

Theil-Sen 125 - 27 - 61 - 37

Hollander 113 - 53 - 101 - 50

Rao-Gore 122 - 26 - 57 - 7

Classical 128 55 51 17

The first row and first column give the emcpirical power
times 1000. Entries in the remainder of the table are
expressed as the difference from the corresponding Sievers-
Scholz result, with a negative value indicating a lower
power than Sievers-Scholz.

Table 15. Empirical Power Times 1000 Under Design C for the
Normal Distribution (a=.125).

^1-2

.218

.436

.654

Sievers-Scholz

112

371

687

918

Theil-Sen

117

- 17

- 48

- 35

Hollander

110

- 52

- 88

- 91

Rao-Gore

103

- 30

- 49

- 44

Classical

117

6

58

36

The first row and first column give the empirical power
times 1000. Entries in the remainder of the table are
expressed as the difference from the corresponding Sievers-
Scholz result, with a negative value indicating a 'lower
power than Sievers-Scholz.

91

Table 16. Empirical Pcv/er Times 1000 Under Design C for the
Double Exponential Distribution (a=.125).

^1-2

.218

.436

.654

Sievers-Scholz

110

354

612

854

Theil-Sen

117

- 16

- 20

47

Hollander

123

- 46

- 41

- 65

Rao-Gore

113

- 25

- 26

34

Classical

111

23

61

37

The first row and first column give the empirical power
times 1000. Entries in the remainder of the table are
expressed as the difference from the corresponding Sievers-
Scholz result, with a negative value indicating a lower
power than Sievers-Scholz .

Table 17. Empirical Power Times 1000 Under Design C .f or the
Cauchy Distribution (a=.125).

Sievers-Scholz

131

320

465

574

Theil-Sen

122

- 13

9

1

Hollander

132

- 57

- 50

- 41

Rao-Gore

125

- 42

- 25

- 11

Classical

141

- 23

2

4

The first row and first column give the empirical power
times 1000. Entries in the remainder of the table are
expressed as the difference from the corresponding Sievers-
Scholz result, with a negative value indicating a lower
power than Sievers-Scholz.

'-n^TP'lV ^'/f-r^ t^*K-j* ^fr w^-;

92

Tables 18, 19, 20, and 21 present these same results for
design D. Recall that designs C and D consisted of samples
of size 6 and 8 per line, respectively, and the exact
nonparametric tests were used under these designs. The
relative performance of the tests using these small sample
sizes is similar to that of the approximate tests under
larger samples, except that the dominance of the classical
t-test is more dramatic under the smaller sample sizes.
Only for Cauchy errors under design D does the power of the
Sievers-Scholz test clearly dominate the classical test.
The Sievers-Scholz and Theil-Sen tests generally exhibited
greater powers than the Hollander and Rao-Gore tests. The
one exception to this occurred under uniform errors, where
the observed power of the Rao-Gore test was slightly higher
than that of the Theil-Sen test, although we note that in
these cases the Sievers-Scholz test had highest power among
these three tests.

For design E, Tables 22 and 23 present empirical levels
and powers for the normal and Cauchy distributions, respec-
tively. When compared with the Theil-Sen test, the
approximate Sievers-Scholz test had higher power under
normal errors and lower power under Cauchy errors. Hence
these results do not indicate a clear choice between these
two methods. However, the Sievers-Scholz and Theil-Sen
tests once again exhibit greater powers than the Hollander
and Rao-Gore tests (except in one instance v/here the power
of the Rao-Gore test marginally exceeded that of the

1 »:;»J Tl C i»i« Tll, ■I JM"'i ifrT°"

93

Table 18. Empirical Power Times 1000 Under Design D for the
Uniform Distribution (c=.057).

Sievers-Scholz

059

300

661

Theil-Sen

064

- 26

- 36

Hollander

056

- 68

- 160

Rao-Gore

064

- 16

- 25

Classical

057

31

81

The first row and first column give the empirical power
times 1000. Entries in the remainder of the table are
expressed as the difference from the corresponding Sievers-
Scholz result, V7ith a negative value indicating a lower
power than Sievers-Scholz.

Table 19. Empirical Power Times 1000 Under Design D for the
Normal Distribution (a=.057).

^1-2

.186

.372

Sievers-Scholz

Theil-Sen

Hollander

Rao-Gore

Classical

059
056
050
054
058

223
9

- 41

- 28
34

551

- 41

- 120

- 47
94

The first row and first column give the empirical power
times 1000. Entries in the remainder of the table are
expressed as the difference from the corresponding Sievers-
Scholz result, with a negative value indicating a lower
power than Sievers-Scholz.

=»(i««jicsc;;i— ST*Mfii

94

Table 20. Empirical Power Times 1000 Unc3er Design D for the
Double Exponential Distribution (a=.057).

1-2

Sievers-Scholz

Theil-Sen

Hollander

Rao-Gore

Classical

.372

.558

059

451

684

054

- 11

- 23

053

- 92

- 135

059

- 22

- 29

063

60

90

The first row and first column give the em.pirical power
times 1000. Entries in the remainder of the table are
expressed as the difference from the corresponding Sievers-
Scholz result, with a negative value indicating a lower
power than Sievers-Scholz.

Table 21. Empirical Power Times 1000 Under Design D for the
Cauchy Distribution {a=.057) .

^1-2

.372

.930

Sievers-Scholz

Theil-Sen

Hollander

Rao-Gore

Classical

058
057
064
057
062

342

- 17

- 71

- 28

- 65

645
17

- 112

- 63

- 44

The first row and first column give the empirical power
times 1000. Entries in the remainder of the table are
expressed as the difference from the corresponding Sievers-
Scholz result, with a negative value indicating a lower
power than Sievers-Scholz.

g i Q Pi gaog i. s^j!yit*giai

95

Table 22. Empirical Power Times 1000 Under Design E for the
Normal Distribution (a=.047).

'1-2

100

200

Sievers-Scholz
Theil-Sen
Hollander
Rao -Go re
Classical

058

228

576

064

- 18

- 48

056

- 46

- 98

058

- 46

- 46

062

- 14

46

1) The first row and first column give the empirical power
times 1000. Entries in the remainder of the table are
expressed as the difference from the corresponding
Sievers-Scholz result, v;ith a negative value indicating a
lower power than Sievers-Scholz.

2) All powers based on 500 simulations.

3) An approximation to the exact Sievers-Scholz technique
was used by selecting a random sample of 10,000 permuta-
tions of the rank vector.

Table 23. Empirical Power Times 1000 Under Design E for the
Cauchy Distribution (a=.047) .

1-2

.100

.200

Sievers-Scholz
-Theil-Sen
Hollander
Rao-Gore
Classical

060

146

288

062

2

36

084

- 34

- 72

046

- 12

- 18

054

- 32

- 96

1) The first row and first column give the empirical power
times 1000. Entries in the remainder of the table are
expressed as the difference from the corresponding
Sievers-Scholz result, with a negative value indicating a
lower power than Sievers-Scholz.

2) All powers based on 500 simulations.

3) An approximation to the exact Sievers-Scholz technique
was used by selecting a random sample of 10,000 permuta-
tions of the rank vector.

96

Theil-Sen test) . Again the classical t-test performs poorly
under Cauchy errors and generally best under normal errors,
although at 6-,_2 = .100 under normal errors, the approximate
Sievers-Scholz test had the highest pov/er.

Summary of Conclusions

The Monte Carlo results presented here generally concur
with the asymptotic relative efficiencies (AREs) discussed
in Section 2.4. Hence we draw the following conclusions
about the power of the tests compared in the two line
setting assuming common regression constants. First con-
sider comparisons of the Sievers-Scholz, Hollander, and
Rao-Gore methods, all three of which include exact confi-
dence intervals for the slope difference g, „. The AREs and
Monte Carlo results indicate superiority of the Sievers-
Scholz approach when compared with the other two exact
nonparametric procedures. Thus we recommend the use of the
proposed application of the Sievers-Scholz approach over the
Hollander and Rao-Gore methods.

Mow consider comparisons of the Sievers-Scholz test.
Sen's test, and the classical least squares theory t-test.
As rank tests both the Sievers-Scholz and Sen test can be
expected to be robust to gross errors, an advantage these
two tests hold over the classical t-test. Although Sen's
test has higher ARE than the Sievers-Scholz test under
heavily tailed distributions, recall that Sen's test

— ■T-.T ^v»> J . , ■-. ,i,, n y,— ,— >^<»^,^.,n^^Trs^-7-r**>— -gsr ,i , ii>, P ->. Tn .i m -W, »iA..- »; '>f^»wv*'-<«V

97

is distribution-free only asymptotically and does not
include exact confidence intervals for the slope difference,
^1-2* ^^ addition, Monte Carlo studies by Smit (1979) and
Lo, Simkin, and Worthley (1978) indicate the power of Sen's
test under small sample sizes is quite conservative with
respect to the corresponding classical least squares test.
Hence the Sievers-Scholz method is preferred over Sen's
technique when an exact, distribution-free test is desired
that includes associated exact confidence intervals for the
slope difference, &^_2- Although the classical test has the
highest overall power when the underlying errors are normal,
choice of the Sievers-Scholz test is appropriate when an
exact test is desired that maintains good power over a range
of distributions, including heavily tailed ones such as the
Cauchy distribution.

Our Monte Carlo comparisons show the Sievers-Scholz
approach generally has greater power than the Theil-Sen
technique when the regression constants are unequally
spaced. When the approximate versions of these two tests
are being used under large samples, the computations
required by each are about the same. Hence in this case we ■
recommend the Sievers-Scholz technique over the Theil-Sen
approach. When exact versions of the tests are being
applied under sample sizes small enough to allow computation
of the exact null distribution of the Sievers-Scholz
statistic (8 or fewer observations per line) , we again

98

recommend the Sievers-Scholz technique over the Theil-Sen
approach. For larger sample sizes when the exact Sievers-
Scholz test is not feasible, we recomm.end the Theil-Sen test
if an exact distribution-free test is desired. The portion
of our Monte Carlo study that included the Sievers-Scholz
approximation based on random samples of permutations has
limited scope and produced mixed results. Thus, further
investigation of this technique is required before a judg-
ment of its usefulness can be made.

CHAPTER THREE
COMPARING THE SLOPES OF SEVERAL LINES

3.1 Introduction
In this chapter we consider the case of several regres-
sion lines, assuming coimnon regression constants for all
lines. We examine the situation where one of the lines is a
standard, or control, with which all other lines are com-
pared. Suppose there are k lines and assume, without loss
of generality, that the kth line is the standard. The
specific topic of this chapter is the comparison of the
slopes of the first k-1 lines with the slope of the kth
line.

As an example of the situation we are considering,
suppose a dentist is examining some measure of strength for
several different experimental amalgams used to fill cavi-
ties in teeth. He may wish to compare the strength of each
of these amalgams with the filling material he has commonly
used in the past. His standard, then, is the commonly used
material. Suppose he measures the strength of samples of
each material, including the standard, after soaking in
water for 2, 3, 4, and 5 days. The strength of an amalgam
will generally decrease with increased soaking time. For
each amalgam., consider a simple linear regression of the

99

100

response, strength, on the independent variable, soaking
time. The slope of each regression line is a measure of the
rate at which the amalgam's strength decreases v/ith
increased soaking time. The methods suggested in this
chapter apply when the dentist wishes to compare the slope
of the line resulting from each experimental amalgam with
the slope of the line resulting from the standard filling
material.

In the rest of this section we formally establish the
linear model assumed and the null and alternative hypotheses
under consideration. We propose a statistic applicable in
this setting whose form follows as a generalization of the
statistic T(b) (2.1.5), discussed in Chapter Two. In
Section 3.2 we derive asym.ptotic distributions of the
proposed statistic, and specify a test of the null hypo-
thesis based on the statistic. In Section 3.3 we illustrate
how our proposed statistic can be used to conduct exact
tests of the null hypothesis. A competitor to the proposed
test is defined and explained in Section 3.4. We close this
chapter in Section 3,5 v/ith Pitman asymptotic relative
efficiencies comparing the proposed test with its com.peti-
tor .

We now establish the linear model assumed throughout
this chapter. Let

Y . . = a, . + e .Y. . + E. . , f31M

101

where i = 1, 2, ..., k, j = 1, ..., N, and

^2 — ^2 — ' * ' - ^N * "*"" this model a . and g., i = 1, ,.., k,
are unknown regression parameters, the x's are known
regression constants, and the E's are mutually independent,
unobservable random errors. We wish to test the null
hypothesis

-0* ^1 " ^2 " ••• " ^k (3.1.2)

against the alternative

-1* ^i ^ ^k ^°^ ^^ least (3.1.3)

one i, i = 1 , . . . , k-1 .

Thus the null hypothesis (3.1.2) is that all k lines are
parallel. Note that this hypothesis can be expressed in
terms of the slope differences, B- -B,, i=l, ..., k-1:

So= ^1 - ^k = ^2 - \ = ••• = ^k-l - ^k = °

Define

?

^i: = ^ij -^kj'

i - 1, ..., k-1, j - 1, ..., N. In terms of the underlying
model parameters.

102

^ij = ("i-°'k) + (^i-^k^^j + (^ij-\j)' (3.1.4)

and so for fixed i, the differences Z. ., i = i n
follow a simple (single line) linear regression model. The
case k = 2 (i=l) was the topic of Chapter Two of this
dissertation. Recall in that case our proposed statistic,
T(b) (2.1.5), results from the use of a statistic originally
suggested by Sievers (1978) and Scholz (1977) for the single
line setting, when applied to the differences Z. ., i = l
..., N. Analogously, here we define for i = 1, ..., k-1

T. (b.) = (1/N) ZZ X sgn(Z. -Z . -b . x ), (3 15)
X 1 ^^^ rs ^ 'is ir 1 rs ' k^.x.o)

where b^ are constants, x^^ = x^ - x^, and sgn ( • ) is as used
in (2.1.2). Note that T^ (b) is T(b) (2.1.5) of Chapter Two
when only considering the ith and kth lines. Thus from the
results in Section 2.3 and 2.5, T^(0) can be used to test
^Oi' ^i " ^k ag^i^st H^^: 3j_ ¥^ &^. Let

2 N - 2
^x = .^ (^i"^) /N (3.1.6)

j=l -J

and

4, A = N'^x/^' (3.1.7)

as in Chapter Two.

103

By Theorem 2.2.1 it follows that under H

Oi'

'i ~ ^k ^"■'^

certain regularity conditions, the asymptotic (II^-) distri-
bution of

T.(0)

T,A

is standard normal. Note also that

k-1
Q^0i=20'

(3.1.8)

that is, Hq holds if and only if h^^ holds for all i = 1,

..., k-1. The asymptotic* normality of each T.{0) under H

X oi

and the relationship C3.1.8) between the H^^ and H suggest
a possible approach to constructing a statistic for testing
Hq using the T^(b^), i = l, ..., k-1:
Let

T =

-1
^T,A

T^{0)
T2{0)

Vl(0)

(3.1.9)

Suppose we can show that under H^ , the asymptotic distribu-
tion of T is multivariate normal with mean vector (zeros)
and nonsingular (k-1) by (k-1) variance-covariance m.atrix

104

i;.. Denote, the inverse of |; by | . Then it follows from
results in Serf ling (1980, p. 25, 128) that the asymptotic
(N-^°°) distribution of

S = T ^"^ T (3.1.10)

under H^ is central chi-squared with k-1 degrees of freedom.
This asymptotic distribution could be used to construct an
approximate level a test of H^ based on S. Our purpose here
is to motivate the use of S (3.1.10), a suitable quadratic
form in the elements of T, to test H . In the next section
it will be shown that S does have an asymptotic chi-squared
distribution under given conditions. The specific form of f
and the test of H^ (3.1.2) against H^ (3.1.3) will be given,
along with the distribution of S under a certain sequence of
alternatives.

3.2 Asymptotic Theory and a Proposed Test
In this section we develop the necessary asymptotic
theory to derive a test of H (3.1.2) versus H (3.1.3)
based on a suitable quadratic form in the elements of T
(3.1.9). We also give the asymptotic distribution of T
under a certain sequence of alternatives. This distribution
will be needed in Section 3.5, when deriving asymptotic
relative efficiencies.

We first give some notation that will be needed in this
section and the rest of Chapter Three. Some notation from

"l'"*»W^VU'»i»*,*:t^.-i.«» -I

105

i

Chapter Two will be repeated here, for the convenience of

Basic notation . Let ^ indicate convergence in proba-
bility and _^ indicate convergence in distribution. Let

2 N - 2
a^ = Z (x.-x) /N, (3.2.1)

j=l -"

2 N - 2 2
""t A " ^ (x.-x) /3 = Na;/3, (3.2,2)

' i=l -^

s-1
^"s " .^ ^^s"^j^ ' (3.2.3)

N
X • = Z (x.-x ) , and so (3.2.4)

j=r+l -■ ^

x.j - Xj. = N(Xj-x) (3.2.5)

by simple algebra. If the event A occurs with probability
one we write A, w.p. 1. We adopt the 0(«), o(«), (•),
o (•), and ^ notations as in Serfling (1980, p. 1, 8-9).

Matrix notation . Some matrix notation v/ill be needed,
A vector of size n is considered to be an n by 1 matrix (a
column vector) . Two such vectors that will be needed are

106

X =

(3.2.6)

n

a vector of constants used in forming linear combinations,
and

=

(3.2.7)

the vector of zeros. The dimensions will be implied by the
context. Also let

^k =

1

• • •

1

• • •

1

• • •

... 1

(3.2.8)

the identity matrix of order k, and

^k =

1 1 1 .. . 1
1 1 1 ... 1

1 1 1 ... 1

(3.2.9)

107

the k by k matrix whose elements are all one (1) . When
multiplication of a matrix by a scalar is indicated, all
elements of the m.atrix are multiplied by the scalar, i.e..

5J3 =

5 5 5
5 5 5
5 5 5

The inverse of a matrix, if it exists, is indicated by a
superscript -1, i.e., I"""- | = I^^. Finally, X' indicates the
transpose of the vector _A.'

Distributional notation . We assxime the underlying
error distribution of all lines is the same, with cdf F and
pdf f. Let E . , j = 1, 2, ..., designate independent and
identically distributed random variables with cdf F. G and
g denote, respectively, the cdf and pdf of E - E . The cdf
and pdf of E^ + E2 - E- are denoted by M and m,
respectively. Let

A(F) = P{E^<E2+E3-E^, E^<E2+Eg-E^} - 1/4

(3.2.10)

and note that A (F) may also be written

A{F) = /[l-M(e)]''dF(e) - 1/4

(3.2.11)

if the integral exists. Let

108
A* = 12A(F), (3.2.12)

and for general pdf q let

Kq) = /q^(x)dx, (3.2.13)

provided the integral exists. For the multivariate normal
distribution of dimension k with mean vector u and
covariance matrix ^ we use the notation

N^^ii'l)- (3.2.14)

For the chi-squared distribution with k degrees of freedom
and noncentrality parameter 6 we use the notation

2
X (<S)- (3.2.15)

Hypotheses . The following hypotheses will be referred
to frequently:

-0' ^1 " ^2 " ••• " ^k (3.2.16)

-1' ^i ^ ^k ^°^ ^^ least one i, (3.2.17)
i = 1, . . . , k-1

^Oi= ^i = ^k (3.2.18)

^li= ^i ^ ^k (3.2.19)

109

Hj,: B. = Bj^ + ((../(Nk)^a^) (3.2.20)

For expectation under H^ we use E .

Conditions. The following set of conditions will be
used in the statements of the theorems in this section:

^ - 2 2
I. I (x.-x) = Na -> ~ as N ^ cc.

j = l ^ ^

II. max (x.-x)^ max (x.-x)^

l^J^N ^ 1^<N ^

^iq^^ - — 2 ->• as N >

2 (x.-x) 2 Na^
j=l ^

III. G is continuous.

IV. G has a square integrable absolutely
continuous density g.

We now prove that the asymptotic distribution of T
(3.1.9) is multivariate normal, as conjectured in the
previous section. We first state and prove two lemmas.

Lemma 3.2.1; Let {X^, n ^ 1 } be independent bounded
random variables with

X

nl 1 ^n' ^'P* ^ for n ^ 1. (3.2.21)

— - •!*— ^ ■ii»»,'>j1i»|t^— - ■^f^..

110

^ 2 ^ 9

Let S^ = IX. and s = I E (X -EX ) ^ n > 1, and assxme
3=1 J " j=i J J

2
s^^ -> CO as n -^ CO. If the bounds B satisfy

^n = °^^n^ (3.2.22)

then

S - ES

i ^ . N(0,1). (3.2.23)

n

Proof of lemma ;

^^^ ^n = ^n ~ ^^n- "^^^^ ^<^n^ = °' ^^^

^^ ^ j!l^^''j"'=''j^^= -Ei^^i^- (3.2.24)

In view of Liapounov's corollary (Chow and Tiecher, 1978,
p. 293) to the Lindeberg-Feller central limit theorem, the
result (3.2,23) follows if we can show the following
conditions hold:

For some 6 > ,

1) E|Yj2+fi <„,„>! (3.2.25)

2) ,^_^E|Yj| ^ = o^^n"^"^) as n -^ -. (3.2.26)

Ill

We will show (3.2.25) and (3.2.26) hold for 6=1. From

(3.2.24) we see eIy P < E(2B )^ = 8B"^ < «, n > 1, and so

n ' — n n —

(3.2.25) holds for 6=1. Now

z e|y.|^ i: e(|y |2|y I)
j=l -' ^ 2^

So

3 3

n n

Z^E(|Yj| 2B.) (^^^^ (3.2.21))

n
2 max B-. Z E( |y. | )

n

2 max B .
s

(from (3.2.24) ) .

n

E E I Y . I 2 max B .
izl_i_ < _IlJin_2 , (3.2.27)

If V7e can show

■ umnii 'i L i ii i iL-j ii jiM«j i i. i M ii j i» . i m f tf « »j i n ■! Il l »i »ii«a »» ir

112

2 max B .

n

(3.2,28)

as n ^ «, then the Liapounov condition (3.2.26) holds for
5 = 1 and the proof is complete. In the following let m be
an integer such that m + 1 < n:

max B

max B . max B .
l^i^n ^ l:£i^m ^ _^ m+l^i^n ^

n n n

max B .
^ l^j^m -* , , , .

1 s ^ ^^^ (B./s.).

n m+l<j<n -' -^

(3.2.29)

Hold m fixed and let n -> «> in (3.2.29)

max B .

lira i£^lIL_ < + sup (B./s.).
n-*-" n j>m+l -' -^

(3.2.30)

Since the left-hand side of (3.2.30) does not involve m, we
let m -> »:

max B .

lim — !££ — = lim sup (B./s.) = 0,

n->-<» n m>°° j>m+l -' ^

(3.2.31)

where the right-hand side of (3.2.31) equals zero because
Bj = o(Sj) (3.2.22). So (3-2.28) is true and the proof is
complete.

113

Leirana 3.2.2 ; Define for i = 1, , k-1.

N

T. (0)

,^ E [T (0)|E -E ],
= 1 -0 ^ ^3 ^3

(3.2.32)

where T^(0) is T^(.b^) in (3.1.5) with b. set to zero. Let

T = a

-1
T,A

T^(0)
T2(0)

(3.2.33)

L

Tj,_,(0)

Under Iq* ^1 " ^2 " *
III,

- 3j^, assuming conditions I, II, and

i t \^i^9.4)

(3.2.34)

as N ->■ "^ where

t = Vi + A* (\.r\-i^'

(3.2.35)

and A* is given by (3.2.12).

114

Proof of lemma :

From (3.1.4) it follows that under H^ ,

Z. -Z. =E. -E, -E. +E, ,
IS ir IS ks ir kr

Then

E„ [T. (0) E. .-E, .]
H^ 1 ' ij kj-"

= E„ [4 SZx sgn(Z. -Z . ) Ie. .-E, .]

H„'N rs ^ 'is ir' ' 11 ki-'

—0 r<s J -I

E[rz izx sgn(E. -E, -E. +E, ) Ie. .-E, .]
N rs ^ 'is ks ir kr' ' ig k:]-"

= ^^ZZx^3E[sgn(E.3-E^3-E.^+E^^) |E. ^-Ej^.] . (3.2.36)

For notational simplicity let D = E. .- E, . . Then examining

13 K^]

E[sgn(E.^-E^^-E.^+Ej^P |e. j-E,^..D] , (3.2.37)

note the following:

1) If s = j then D is independent of E. - E, and so
(3.2.37) equals

115

P{D>E, -E, } - P{D<E. -E, }
ir kr xr kr"^

= G(D) - {l-G(D))

= 2G(D) - 1. (3.2.38)

2) Similarly, if r = j, then (3.2.37) equals

1 - 2G{D) . (3.2.39)

3) If r 7^ j and s ?^ j , then D is independent of
^is " ^ks " ^ir ■*■ ^kr ^^'^ ^° (3.2.37) equals

E[sgn(E.^-Ej^^-E.^+Ej^p]. (3.2.40)

Now E^^ - Ej^g - E^ + E, is symmetric about zero so
(3.2.40) equals zero.
Recall the notation x. (3.2.3) and x . (3.2.4).
Applying 1), 2), and 3) above to (3.2.36) results in the
following:

%tT.(0)lE..-E3^.]

= TT ZZx E[sgn(E. -E, -E . +E, ) Ie. .-E, .1
N _^ rs ^ ' IS ks ir kr' ' 13 k^

116

2 J-1 N

n[ r X (2G(D)-1) + r x(l-2G(D))]

r=l J s=j+l -"^

|[{2G(D)-l)x_ + (l-2G(D))x..]

^(2G(D)-1) (x.^-x^.)

= ^(2G(D)-l)N(x.-x) (from (3.2.5))

= (2G(D)-l)(x.-x). (3.2.41)

Substitute (3.2.41) into (3.2.32):

N

T,(0) = fjE2^[Ti(0)|E..-E^.]

N
= T (2G(E -E )-l) (x.-x)

N

Z_^2(Xj-x)G(E^j-Ej^j). (3.2.42)

We have now established the form of the T. (0), i = 1,
k-1. Consider an arbitrary linear combination of the
i\(0) -s.

117

k-1 .

.f/i^i^O). (3.2.43)

Our basic approach to proving this lemma is to show the
asymptotic normality of this linear combination (3.2.43).
Let

k-1
Uj = 2(x.-5)_Z A.G(E -Ej^n. (3.2.44)

i=l

From (3.2.42) ,

k-1 . k-1 N

J^A,T,(0, - ,f^VJ^2(x.-5,G(E,.-E,.,

N _ k-1

= Z [2 (x.-x) Z A.G(E. .-E, .) ]
j=l 3 i^i 1 13 k:' J

N
■ l^^r (3.2.45)

Note that U^ and U^ , are independent, j 7^ j ' . Before
deriving the mean and variance of U , observe the following:
1) The cdf of E^j - E^. is G (i<k) . Let g"^ be an

inverse of G, i.e., G"^(G(t)) = t , <_ t ^ 1 . G~^

exists by continuity of G (condition III) . Then
the cdf of G(E^j-E^.) is given by

-.--.-.-.^^1.3=

118

P{G(E..-Ej^.)<t}

Pi'E^^-E^.<C~^ it) }

G(G ^(t))

= t, < t < 1.

Thus G(E^j-Ej^. ) is uniformly distributed over
[0,1] . It follows that

E[G(E. .-Ej^.)] = 1/2, (3.2.46)

and

E[G(E. .-Ej^.)-l/2]2 = 1/12, (3.2.47)

the variance of a uniform [0,1] random variable.

2) Recalling the definition of A (F) (3.2.10), it
follows that

A(F) + - = P{E2<E^+E^-E2, E2<E3+Eg-E^}

= E tP{E5-E4<E^-E2 ,E^-Eg<E3-E2 | E^=e^,E^=e^,E^=e^}]

119

= E[P{E5-E^<e^-e2,E^-E^<e3-e2|E^=e^,E2=e2,E3=e3}]

PiE^-E^<e^-e^\E^=e^,E^=e^,E^=e^}]

= E[G(E^-E2)G(E3-E2)] ,

and so using (3.2.46) we have

E[{G(E^-E2)-l/2) (G(E3-E2)-l/2)] = A (F) . (3.2.48)

Then

k-1
EU. = E[2(x.-x) Z A.G(E. .-E, .)]

= 2(x.-5)_Z^A.E[G(E..-E3^.)]

k-1

(Xj-x)_z X^ (from (3.2.46)), (3.2.49)

and

120

Var(U.) = E[U.-EU.] ^

k-1 ^

= E[(2(x-x) Z X G(E -E ))-((x,-x) I A.)]
-» i=i -^ -^J ^-J J j__2 1

k-1
= E[2{x.-x) I X. (G(E.j-Ej^.)-l/2)]2

- 2 '"-I 2
= 4{x -X) { I X^E[G(E.,-E, .)-l/2]

+ 2,2Z^A.X.,E[{G(E. .-Ej^.)-l/2) (G (E . , ,-Ej^ . ) -1/2) ] }

- 2 ^-1 P
= 4(x -X) [(1/12) E A^ + A(F)2 ZI A.A.,], (3.2.50)

i=l i<i' ^ ^

where (3.2.47) and (3.2.48) were applied to arrive at the
final expression (3.2.50). Now we show that the conditions
of Lemma 3.2.1 apply to the U.. First note

lu^l = I2(x^-5)_I^A.G(E.^-E^^)|

< 2 max ix -x| z | A | , w.p. 1
l<j<N -' i=i ^

121

k-1

Let B = 2 max |x -x| z |^ |, and then

l<j<N -' i=l ^

U^l 1 B^, w.p. 1, N > 1, (3.2.51)

SO (3.2.21) of Lemma 3.2.1 is satisfied. Applying (3.2.50),

2 ^
if we let s = z Var(U.), then

j=l ^

2 ^ - 2 ^-^7
s^ = 4 S (X -x)^[(l/12) Z Af+A(F)2 ZZ A.A.,]

j=l ^ i=l ^ i<i' 1 ^

N _ k-1 _
= (1/3) Z (x,-x)^[ Z Xf+2A* ZZ \.X.,]. (3 2 52)

j=l ^ i=l ^ i<i- 1 1 U.^.i^)

For later reference we note:

4 = Varjj^(ZA.T.(0)), (3.2.53)

and taking A. = 1, a^, =0, ± f x' the form of the null
variance of T^(0) follows from (3.2.52):

r.2 2

Var_^_^(T.(0)) = (1/3) Z^(x_.-x)^ = a^^^. (3.2.54)

122

Now examine B„/s :
N N

k-1
max |x.-x|2 Z \x.
B^/Sm = l^i^N -J i=l ^

{(1/3) Z (X -x)'^[ Z X^+2A* EZ X.X.,]}^
j=l ^ i=l ^ i<i' 1 1

- as N - " (3.2.55)

by condition II, and so (3.2.22) of Lemma 3.2.1 holds. We
note from (3.2.49) ,

N N k-1
2 EU. = Z (x.-x) Z A. =
j=l ^ j=l 3 ,i=i 1

Applying Lemma 3.2.1,

N N

2 U . - Z EU. ^

1=1 ^ j=i : d

-^ i ^-^^— ^ N(0,1)

N

as N ->■ » ,

N

Z U. ^

-^i > N(0,1) (from (3.2.56)),

(3.2.56)

123

:>

k-1 .

Z A.T. (0)
i=l d

=>

N(0,1) (see(3.2.45) ) ,

N

_1 k-1 .
a Z XT. (0)

i=l d
^I ;N(0,1), (3.2.57)

as N ->• "

Let A be a vector as in (3.2.6) and recall the definition of
the matrix | (3.2.35). From (3.2.52) the following
expression results:

,-2 2 , -2

N

- 2

''t^A^N = f^T,A(^/3) Z (X -x)^]A'tX

j=l -^

= ^^tVt,a^A'^A

= A'iA.

(3.2.58)

Using (3.2.58) in (3.2.57) we see

=%fN(0,l) (3.2.59)

A'l d

ix'tx)

as N -^ CO, where T is given by (3.2.33). The result (3.2.59)
essentially shows that any linear combination of the

124

elements of T converges in distribution to a random variable
with the distribution of the same linear combination of a
\_l(0.4) random variable. By a theorem in Serfling (1980,
p. 18) this is equivalent to showing T has an asymptotic
(N->") Nj^_;l^^'^^ distribution, and the proof of Lemma 3.2.2
is complete.

Theorem 3.2.3: Under H^ : 3^ = 33 = . . . = 6j^, assuming
conditions I, li, and III,

d

"k-l^-' + '' (3.2.60)

T " N,,_^(0,t)

as N -. -, where T is given by (3.1.9) and t is given by
(3.2.35).

Proof ;

Consider f or i = 1 , ..., k-1,

E„ (T. (0)-T. (0))^,
-0 ^

Where T^(0) is given by (3.2.32). T^ (0) is called the

projection of T. (0) on the Z..=E..-E i=l n

under H^ . By a lemma due to Hajek and appearing in Serfling

(1980, p. 300), it follows from the form of T. (0) that

E (T.(O)-T. (0))2 = var^ (T.(0)) -Var^ (T, (0)),
u —0 —0

(3.2.61)

125

1, ..., k-1. From (3.2.54),

Var^^ (T^(0)) = aj^^, (3.2.62;

and from Theorem 2.2.1 applied to this situation.

Varjj (T^(0)) = a^^^ + o^. (3.2.63)

Substituting (3.2,62) and (3.2.63) into (3.2.61) yields

E (T.(0)-T.(0))2 = ia^^-^ol) - a^^ = o^

■■> lim a \e (T. (0)-T (0))2 = lim a^ o^

N - 2

2 (Xj-X)^

j=l ^

= 0, (3.2.64)

for i = 1 , — , k-1 . Then it follows that

crT^^^T^(O) - T^(0)) I (3.2.65)

as N ^ ", i = 1, ..., k-1. This implies

126

_l k-1 k-1

^T^A^^^i^i^O) - ^ ^iTi(O)) P (3.2.66)

1=1 i=i -^ -^

as N ^ oo. Using the matrix notation (3.2.6), (3.2.33), and
(3.1.9), we can express (3.2.66) as follows:

X'T - X'T P

(3.2.67)

as N ^ 00. Thus from Slutzky's theorem (Randies and Wolfe,
1979, p. 424), (3.2.67) implies that if X' T has a limiting
distribution then X'T has this same limiting distribution.
In Lemma 3.2.2 we saw that under H^ , assuming conditions I,
II, and III,

X ' T

~ ~ t: t N(0,l)

ix'tx)

as N -> 00 (from (3.2.59)), and so

k . N(0,1) (3.2.68)

iX'iX)

as N ^ 00. As argued in Lemma 3.2.2, this is equivalent
(Serf ling, 1980, p. 18) to showing T has an asymptotic (N->oo)
N]^-l(£'t) distribution, and the proof of Theorem 3.2.3 is
complete.

127

Notice that | (3.2.35) has the form

abb
b a b
b b a

b b b .

b
b
b

(3.2.69)

with a = 1 and b = a* = 12A (F) . Then by Theorem 8.3.4 in
Graybill (1969, p. 171), | has an inverse if and only if
(iff)

1 7^ 12A(F) ,

(3.2.70)

and

1 7^ -(k-2)12A(F) .

(3.2.71)

Mann and Pirie (1982) give the following bounds for A (F) :

1/36 < A(F) < 1/24.

(3.2.72)

The bounds (3.2.72) hold for all continuous distributions.
Condition (3.2.70) fails iff a (F) = 1/12, which, in view of
(3.2.72), is not possible for continuous F. Condition

(3.2.71) fails iff A(F) = -[12(k-2)]"^ and k ^ 3. By

(3.2.72) we see that A (F) cannot be negative, and so

128

(3.2.71) holds for continuous F. Hence for continuous F,
^ has an inverse, i~ . Again appealing to Graybill's (1969,
p. 171) theorem, the inverse of | is given by

t = riA^^^k-l ~ l-(k-2)A* "^k-l^ • (3.2.73)

S = T'f^T, (3.2.74)

T as in (3.1.9). Using (3.2.73) and some algebra, S can be
expressed:

k-1 ^ k-1
(l+(k-2)A*) I (T^(0))-A*( S T^(0))^

S= ^^ ^=^^ . (3.2.75)

a^^^a-h*) (l+(k-2)A*)

The following gives the asymptotic distribution of S under

Theorem 3.2.4 : Under H^ : 3^ = g^ = . . . = g^, assuming
conditions I, II, and III,

S ^ X^_i (0) (3.2.76)

as N -» «, where S is given by (3.2.74)

129

Proof ;

From Theorem 3.2.3,

1 J X

as N -»- ", under H^ , where X is a random vector having the
N]^_l(£4) distribution. Then from Corollary 1.7 in Serf ling
(1980, p. 25), it follows that

S = T' Z~"^T J X' t~'''X

as N ^- ", under Hq . Applying a theorem in Serf ling (1980,
p. 128) it follows that

X'T'^X

2

has the Xj^_j^(0) distribution, and the proof is complete.

The result of Theorem 3.2.4 suggests a test of

-0* ^1 " ^2 " ••• " ^k ^^- -1 (3.2.17). Let X^_^ be the cdf
of the central chi-squared distribution with k-1 degrees of
freedom. For < a < 1, let ^^_^ ^ be the upper 100 (1-a)
percentile, that is X(iJ;j^_^ ^) = i _ a. Then from Theorem
3.2.4, the test based on S that rejects H_ (3.2.16) if

^■^'^k-l,a (3.2.77)

is an approximate level a test of H against H (3.2.17)

130

Of course the parameter A (F) depends on the underlying
distribution and so the value of S depends on the underlying
distribution. Consistent estimates of A (F) are discussed by
Mann and Pirie (1982), however such estimates are often
tedious to compute. Consider Table 24, which is adapted
from Mann and Pirie (1982) and shows the value of A (F) to
four decimal places for several common error distributions.

Table 24. Values of A (F) .

F Uniform Normal Logistic Exponential Cauchy Max. Min.

A(F) .0409 .0402 .0398 .0394 .0379 .0417 .0228

Source: Adapted from Mann and Pirie (1982) .

The last two columns of Table 24 give the maximum and

minimum values of A (F) for continuous F, discussed

previously (3.2.72). We see the range of possible values

for A(F) is about 0.02, and all of the values listed remain

within 0.01 of the maximiom. The range of possible values

for A* = 12A(F) is (l/2)-(l/3) = 1/6 or about 0.17 for

continuous F (see (3.2.72)). In view of this small range we

suggest a test based on S which replaces A* by a value A*

c'

chosen to yield a conservative test. We now indicate how to

choose the value of A*:

c

131

Let

k-1
T - I T,(0)/(k-l) (3.2.78)

i=l ^

2 ^-^ - 2
s^ = ^ (T. (0)-T)^/(k-l) , and (3.2.79)

i=l

2 2-2
c^ = s^/ (T) . (3.2.80)

Then using these definitions in the expression for S
(3.2.75) we have

2

S = S(A*) = [(k-DT^/cT^ ] [-!t__ + 1 _ j^

1-A l+(k-2)A

(3.2.81)

Examining the derivative of S(A*) with respect to A*,
S' (A*) , it can be shown that

S' (A*) is >, = ,<

(3.2.82)

(k-2)^ - c^
as A* is >, = ,< —

(k-2)Cy + (k-2)^

Let S(l/3), S(l/2) be the value of S(A*) when A* is set at
it lower and upper limits of 1/3 and 1/2, respectively.

132

Let

r

1/3 if cl > 1"^

T 2

A* =/ ^^^ " ""t , if kzi < ^2 ^ 4(k-2)

"" \ (k-2)c^ + {k-2)'2 k^ - T - (3^^^ J 2

1/2 if c^ < ^ . (3.2.83)

Using (3.2.82) and the upper and lower bounds of A* it
follows from some routine analysis that

min S(A*) = S(A*). (3.2.84)

(1/3) < A* £ (1/2) ^ .

Then if we define S(A*) to be the value of S (3.2.75) using

*
A* = A , it follows from (3.2.84) that the test of H„

(3.2.16) which rejects in favor of H (3.2.17) when

^(^^ ^ Vl,a (3.2.85)

will be conservative. Its true level will be less than or
equal to the nominal level a. However in view of the small
range of A* under continuous F, the degree of conservatism
of the test (3.2.85) should be slight.

Suppose 0)^^, 0^2 ..., 0)^ are given constants such that
ojj^ = 0, and consider the sequence of alternatives.

133

^N =

'i = 3^^ + to^/{Nk) ^a^,

(3.2.86)

1 X y • * • f K I

Let

(0 =

OJ-

0).

^k-1

(3.2.87)

The following theorems give the asymptotic distribution of T
(3.1.9) and S (3.2.74) under H„.

Theorem 3.2.5. Let ^, o), and T be given by (3.2.35),
(3.2.87), and (3.1.9), respectively. Under H (3.2.86),
assuming conditions I-IV,

T t Nj^_i(2(3^)I(g)k-\,|)

(3.2.88)

as N ^ «>.

Proof ;

Let 0)^^ = oj^/N^^a^, i =

of alternatives is H„: g. =

— N 1

1, ..., k. Then the sequence
^k "*" "iN' ^ ^ ^' •••/ k. Let

134

^^-i^N) = ^t]a

Vi (-"k-1 n)

(3.2.89)

As noted in Chapter Two, the limiting distribution of

T = T{_0) under H^ is the same as the limiting distribution

°f I(~%) under H

. = 3, . So assume H- and

"1 ^2
show that T(-i;Jfj) has the desired limiting distribution.

Consider an arbitrary linear combination of the T. (-u. )

1 iN '

k-1

i=l ^ ^ iN'

(3.2.90)

Using the matrix notation (3.2.89) and (3.2.6), we can
express the linear combination (3.2.90) as

-t,aA't(-%)

(3.2.91)

Our basic approach to proving Theorem 3.2.5 is to show this
linear combination has an asymptotic distribution that is
univariate normal. We first state and prove two lemmas.

Lemma 3.2.6 : Assume H
conditions I-IV. Then

^1 =

= B, and

135

%f.!^^i^i(-^N^/^T,A^ - 2(3'^)I(g)k^^E^A.co. (3.2.92)

as N ->- «> .
Proof of lemma ;

Applying Lemma 2.2.3, we have in the notation of this
chapter.

Ejj . fT^(-aj^j^)/a^^^] ^ 2 (3^) I (g) k^u^ (3.2.93)

asN-><=°, i = l, ..., k-1, H_ . given by (3.2.18). Since H-^
implies Hq., i = 1, ..., k-1 (see (3.1.8)), the result
(3.2.92) follows immediately.

Lemma 3.2.7 ; Assume H- ; B- = Bj = ••• = S, and condi-
tions I-IV. Then

.^^i^i(-'^iN) - t.^^i^i^O) -^ V(.^^iTi(-'OiN))]
1=1 1=1 —0 1=1

''t,a

(3.2.94)

as N ->■ ".

Proof of lemma;

Applying Lemma 2.2.4, we have in the notation of this
chapter.

136

^i^-^N^ - f^i^O)-^ V.(Ti (--,,))]

Ol p ^

: (3.2.95)

T,A

asN^co, i = i^ ._^ k_l^ Since H. implies H„ . ,

—0 '^ 1

i = 1, ..., k-1 (see (3.1.8)), the result (3.2.94) follows
immediately.

Proof of Theorem 3.2.5 (continued) ;

Under H^^ and conditions I, II, and III, Theorem 3.2.3
(see (3.2.68) ) showed

X'T J N(0,X'|X) (3.2.96)

as N -> CO, where ^ is given by (3.2.35).

The result of lemma 3.2.6 in matrix notation is

Ejj [A'I(-%)] ^ 2(3'2)I(g)A'iil (3.2.97)

as N -> =0. Using (3.2.96), (3.2.97) and Slutzky's Theorem
(Randies and Wolfe, 1979, p. 424), it follows that

X'T + Ejj U'T(-co^)] J N(2(3'2)I(g)A'iil'i'lA) (3.2.98)

as N ^ =°. The result of lemma 3.2.7 in matrix notation is

^'T(-aij^.) - U'T + Eg (A'T(-aij^))] ^ (3.2.99)

137

as N ^ 00. So using (3.2.98) and (3.2.99), a second

application of Slutzky's theorem shoxvs

^'T(-a)jj) t N(2(3^)I(g)A'i!l/ A'ID (3.2.100)

as N ^ ", under H^ . Put another way, (3.2.100) reveals that
the asymptotic distribution of A'K-w^) under H is the
distribution of ^'X, where X has the Nj^_^ (2 (3^) I (g) k\, ^)
distribution. Hence by a theorem in Serfling (1980, p. 18)
this is equivalent to showing T(-u ) converges in distribu-
tion to this same multivariate normal distribution. Hence
by our remarks at the beginning of the proof of Theorem
3.2.5, the proof is complete.

Theorem 3.2.8 ; Let S be given by (3.2.74). Under H
(3.2.86), assuming conditions I-IV,

k-1 , k-1
(l+(k-2)A*) E a3T-A*( Z to . ) ^

S t xj_i([12l2(g)][ k(l-A*Ml+'(k-2)iM ^ ^) (3.2.101)

as N -»■«>, A* given by (3.2.12)
Proof:

From Theorem 3.2.5, under H„,

— N

1 J X

138

as N -> CO, where X has the Mj^_^ (2 (3^) I (g) k^oj , t ) distribution,
Then from Corollary 1.7 in Serfling (1980, p. 25), it
follows that

S = IT^T f X't ^X (3.2.102)

as N > 00, under H^^. Applying a theorem in Serfling (1980,
p. 128) , X' t~-^X has the

Xj^_^(12l (g)k~^m'i~^w,) (3.2.103)

distribution. The noncentrality parameter in (3.2.101) is
just the one in (3.2.103) expressed in terms of the elements
of 0) (3.2.87) and ^""^ (3.2.73).

3.3 An Exact Test

The statistic S (3.2.75) can be utilized to construct
an exact test of H^ (3.2.16) against H^ (3.2.17). The idea
is similar to that used for exact tests based on T(0),
described in Chapter Two. After first establishing some
notation, we will review the case of k=2 lines. We then
extend this method to the situation of several lines.

Recall the definition

Z. . = Y. . - Y, .

i: ID kj

(a.-aj^) + (6-e^)x, + (E.j-E^j), (3.3.1)

139

where Y^^ follow the linear model (3.1.1), i = 1, ..., k-1,

3=1, ..., N. Let R . . be the rank of Z . , amora

(Z^^, ..., Zj^^}, the Z's resulting froir, the observations at

the ith and kth lines. Consider the vectors

R.
-3

R

Ij
Z

2j

R.

k-1 j

J

(3.3.2)

j - 1, ..., N, and the matrix composed of these vectors,

R = [R^ R2 ... P^] .

(3.3.3)

The following representation of T. (0) in terms of the R^

1 i j '

Z
•••/ Rj,j^T results from (2.5.1):

^ 7
T.(0) = (2/N) 2 Rf X . - (N+l)x

j = l ^^ -'

(3.3.4)

When k=2, R has dimensions 1 by N. In this case
-0 " ^01* ^1 " ^2 ^"^ under H^^ it follows from (3.3.1)
that the Z^^, ..., z^^^^ are i.i.d. and hence the distribution

140

of R is uniform over the N! permutations of its elements,
the integers 1, ..., N. Thus, as discussed in Chapter Two,
the null (6^=62) distribution of T. (0) can be determined by
using (3.3.4) to compute the N! equally likely values of
T.(0).

Now note that when k > 2, the matrix R is (k-1) by N
and the elements within the jth column, R., are dependent,

since their values all rely, in part, on the value of Y . .

ik

To take care of these dependencies, we condition on the
values of the columns of R. That is, under H :

^1 ^ ^2 "^ •*• ^ ^k' ^^^ distribution of R conditional on the
observed values of R^ , j=l, ..., n, is uniform over the N!
permutations of its columns.

The conditional null distribution of R just described
can be used to conduct an exact test of H against H using
S(A^), defined after (3.2.84). The N! values of T (3.1.9)
resulting from this conditional null distribution of R would
be used to compute N! values of S(A*), each occurring under
Hq v/ith conditional probability 1/N! . Suppose, for an
observed set of data, s is one of these values such that

P{S(A*)^s} = a.

where the probability is evaluated using the conditional
null distribution. Then, for this data set, the test which
rejects H^ (3.2.16) in favor of H, (3.2.17) if

141

S(A ) ^ s

is an exact, conditional level a test. Note that the degree
of difficulty of computation of the given permutation
distribution depends only on the number of regression
constants per line, N, and not on the number of lines, k.

o

In view of this fact and the results discussed in

Chapter Two, we found the exact test of H presented here to

be computationally feasible with the use of a computer
whenever N _< 8 .

3.4 A Com.petitor to the Proposed Test
The statistic S (3.2.74) resulted from an obvious
generalization of the proposed application of the Sievers-
Scholz statistic T(0) for use in comparing the slopes of
two regression lines. Clearly one could consider such gen-
eralizations based on the other statistics for testing

— 0* ^1-2 ~ ^1 ~ ^2 "^ '^' P^ssented in Chapter Two. In view
of the fare's reported in the previous chapter, we shall
confine our attention to the statistic defined by Sen
(1969) , keeping in mind that exact tests based on a permu-
tation distribution are not possible for Sen's test
statistic. We would anticipate that, in terms of PARE,
Sen's test is likely to be better than the test based on S.
Recall that the test based on S was designed to test

lio= ^1 = ^2 = ... = 3, (3.4.1)

142

against

-1* ^i "^ ^k ^^^ ^'^ least one i, f3.4.2)

i = 1, . . . , k-1.

Sen proposed his statistic for use in testing H against
the more general alternative,

^1 + ' ^ir •••/ 3j^ are not all equal, (3.4.3)

and hence it would not be appropriate to compare his test

against the test based on S. Instead we modify Sen's

statistic to be specifically sensitive to the alternative H-

(3.4.2). A test based on the modified statistic follows

from the results of Sen. In the next section we compare

this test with the test based on S . We note that the test

based on Sen's results will not require common regression

constants for all lines as required by the test based on S.

However, as in Chapter Two, we will assume this is true to

ease the notation and also to facilitate the comparisons in

the next section. Thus we assume the basic linear model

(3.1.1). The notation defined in Chapter Two when

introducing Sen's statistic in the two line setting will be

repeated here for convenience of the reader.

Let (j) (u) be an absolutely continuous and nondecreasing

function of u: < u < 1, and assume that <^ {u) is square

integrable over (0,1). Let U,,. < U,„. < ... < U ,,,, be the

(-L) (^) (N)

143

iii

order statistics of a sample of size N from a uniform [0,1]
distribution. Then define the following scores:

Ej = E[MU^.j)], (3.4.4)

or

E. = Mj/N+1), (3.4.5)

j = 1 , . . . , N. Define

and

1
(j)* = / (J) (u)du (3.4.6)

A^ = / (()2(u)du - (<j.*)2, (3.4.7)

•

and consider the statistics

'' U

V = [ Z (X-;-^)E ]/(AN^a ) (3.4.8)

j=l ^ ^ij ^

i - 1, 2, ..., k, where R . . is the rank of Y.. among Y.,.

ID 2.J ^ il '

^12' •••' ''^iN' ^^® observations of the ith line. Recall
that statistics such as (3.4.8) are used in the single line
setting to test hypotheses about the slope.

144

Assume F, the underlying error cdf of the assumed model
(3.1.1), is absolutely continuous and

CO

IMF) = / [|^]2dF(x) < <.. (3.4.9)

Then we again reserve the symbol ¥ (u) for the optimal score
function:

T(u) = -^IJlImI , < u < 1. (3.4.10)

f(F ^(u))

It can be shown,

1

/ 'y (u) du = , and

1 2
/ Y (u)du = I* (F)

We also define

P(^,<t>) = [/ 'i'{u)<f (u)du]/[A^I*(F)]^, (3.4 11)

which can be regarded as a measure of the correlation
between the chosen score function cp and the optimal one {"H)

145

I

for the error distribution being considered. The expres-
sions p(T,(j)) (3.4.11) and I* (F) (3.4.9) Vi7ill appear in the
development of the statistic based on Sen's (1969) work.

We now define Sen's statistic in the case of several
regression lines and then give our modification. Let

V^(y^+bx) (3.4.12)

denote the value of V. (3.3.7) based on Y . , + bx, , Y.^ +

1 il 1' i2

bx^, ..., Y.^ + bx . Define

V = r V./k.
i=l

Assuming H^ , let 3* denote the Hodge s-Lehmann estimate of
the common slope of all k lines based on V. Define

^i " ^i(Xi-^*2i) ' (3.4.13)

i = 1, 2, ..., k. Then Sen proposed the statistic

k ^2
L = Z V. (3.4.14)

i=l ^

to test Hq (3.4.1) against H^_^ (3.4.3). The statistic L is
a quadratic form in the V. , i = 1 , . . . , k. We gave an

146

intuitive motivation for the forin of L in Chapter Tvro. We
now define the modification of L.

Let 3| denote the Hodges-Lehmann estimate of 6 . based
on V^ (3.4.8), that is, based only on the observations from
the ith line. Let

Vj_ = V. (Y^-B*x)/ (3.4.15)

i = 1, ..., k, and define

A k -

L = Z (V.-V)2, (3.4.16)

i=l ^

- k .
where V = Z V./k. Since we are interested in rejecting H.
i=l ^

only when the slope of one of the k-1 lines differs from the
slope of the kth line, we propose transforming the observa-
tions by using the estimate of the slope of the kth line,
3*, rather than the pooled slope estimate, 6*. We
again consider a quadratic form, now in the V., i = 1, ...,
k. Thus we have made a reasonable modification in the form
of Sen's original statistic. The basic asymptotic theory
still holds, and proofs leading to the specific form of a
test of H_ against H- based on L will be given.

We now state the basic set of assumptions required for
the results in this section. These are the assumptions Sen

"■^-1 TTf^-nrrar-r-.

147

gives, applied to the setting of common regression
constants:

^ - 2
A. Z (x.-x) •> <» as N ^- <=°.

j=l ^

max (x.-x)

B. l^i^ N ^

— jj -^ as N ^

2 (x.-x) 2

j=l ^

C. F is absolutely continuous with (3.4.9)

I* (F) < 00,

D. The score function <f) (u) is an absolutely
continuous and non-decreasing function of
u, < u < 1, that is square integrable
over (0,1) .

We note that conditions A and B are identical to conditions
I and II, respectively, defined previously. All four of
these conditions will be referred to jointly in the follow-
ing proofs as conditions A-D. Finally, we state again the
sequence of alternatives we consider:

n^: 3. = 63^ + co./N^k^a^ (3.4.17)

-L. X f • * • f }\. f

where 0)^ = 0. Most of the following lemmas are proved in
the work of Sen (196 9) .

■^t-S-,V---W^- .,?Ki^ii

148

Lemma 3.4.1 ; Under H^ (3.4.17) and the conditions

A-D,

N^a^(B^-3|) I = Op(l) (3.4.18)

and

N^0^(6^-S*)| = (1) (3.4.19)

as N •» «>, i = 1, . . . , k.
Proof of leiruna:

The result (3.4.18) is equation (3.20) in Sen (1969,
p. 1675) , when using the notation of this section. Result
(3.4.19) follows from (3.4.18):
Under E^: ^^ = &^ + <^i/N^k^ax,

|N\(6.-e*)

= lN^o^^(6k-3p + a)^/k'^

- I^^'^x^^k-^k^ 1 + Ui/k^

= (1) + |ajj_/k^| (from (3.4.18))

= (1) , as N ^

XT

149

Lenmia 3.4.2 ; Under H (3.4.17) and the conditions
A-D,

|V.(Y.-3|X)| = Op(l) (3.4.20)

as N ^ ", i = 1, . . . , k.
Proof of lemma ;

The result (3.4.20) is equation (3.22) in Sen (1969,
p. 1675) , when using the notation of this section. Taking i
= k, we note (3.4.20) yields:

V.

kl = l\(Ik-^k^H = °p(l) (3.4.21)

as N ■> " . ■

Lemma 3.4.3 ; Let a^ be a positive constant and define
I(aQ) = {a: |a| 1 a^ } . Then assuming conditions A-D,

Vi(Ii-[Bi-a/N^a ]x) - V. (Y . - [ g, -b/N^a ]x)

= p ('l',(j)) (a-b) [I*(F)] + o (1) (3.4.22)

holds simultaneously for all a, b e I(a_) as N

Proof of lemma :

This is proven as Lemma 3.2 in Sen (1969, p. 1674). We
have restated the lemma here in the notation of this
section.

150

Remark ; Sen notes that since (3.4.22) holds
simultaneously for all a, b e i (a^), it follows that
(3.4.22) also holds if a and b are random variables
depending on N and both are (1) as N ^ «.

Lemma 3.4.4: Under H^ (3.4.17) and the conditions
A-D,

^i " ^i^^i " ^k^^ = P('l',<(') [I*{F)]\^a^(6*-e*) + o (1)

(3.4.23)

as N ^ «>, i = 1, . . . , k.
Proof of lemma:

Vi = [V.(Y^-B*x) - V.(Y.-6|x)] + Vi(Y.-6^x). (3.4.24)

Let

a = N^a^(6.-6*) (3.4.25)

and

b = N^c:^(6.-6^). (3.4.26)

By (3.4.19) and (3.4.18), a = (1) and b = O (1) as N > -.

P p

Then in view of the remark following the proof of Lemma
3.4.3 we may apply that lemma with a and b as in (3.4.25)
and (3.4.26), respectively, obtaining

151

Vi(li-ejx) - v3_(Y^-ejx)

= P (^F, (I)) [I*(F) ]\^a^(B^-6*) + o (1) (3.4.27)

as N -> ", i = 1, ..., k. Apply (3.4.27) and (3.4.20) to
(3.4.24) :

Vi = [V^(Y.-e*x) - V^(Y.-B|x)] + V. (Y.-6^x)

= [p(Y,*) [I*(F)]^N'^a^(e^-e*) + o (1)] + o (1)

as N ^ 00, i = 1, _.^ k^ Thus (3.4.23) is verified.

Lemma 3.4.5; Under H^ (3.4.17) and conditions A-D,

^''^x^^i-^k^ f N(kA.,[p2('F,,j,)I*(F)]-l) (3.4.28)

as N -> =0, i = 1, ..., k.
Proof of lemma ;

By Lemma 3.4 in Sen (1969, p. 1676),

N^cr^(e^-e.) f N(O,[p2(*,0)I*(F)]-^) (3.4.29)

as N ->- ". Now under H„,

— N'

- ^T'2

'- 1?. 3^

N^a^(3*-e^-,./N^k^a^)

152

= N^a^(6|-6j^)-(a)^/k^)

, In view of (3.4.29), this implie:

%v d

-1,

N^a^(B|-Bj^) - (oj^/k^) ; N(0,[p''('^,<j,)I*(F)]"^)

as N ■* ", and the desired result (3.4.28) follows
immediately.

Lemma 3.4.6: Let

X

X

X =

— n

2 ,n

X

k,n

X =
— n

.o

•l,n

o

2,n

•

•

.

o

k-l,n_

1 ,n k,n
2 ,n k,n

k-l,n k,n

'

^1

^2

•

p =

•

1

^k

o

u =

'^l-^k

^2-^k

^k-l-^k

(3.4.30)

where X^, n >_ 1 represents a sequence of random vectors and
_M is a vector of constants. Let a > be a constant. If

Xn f ^k^ii'^'V

(3.4.31)

as n ->- <» , then

'■*»»«^ »«W,— -: JSr -i^- -

153

2i° t Nk-i(l°'^^(Vi+Jk_i)) (3.4.32)

as n -*■ °= .
Proof of lemma;

Consider an arbitrary linear combination of the

elements of X°:
— n

k-1 ^ k-1

rO

2 ^^X. ^ = Z A. (X. -X, )
i=l 1 1'^ i=i 1 i.n k,n'

k-1 k-1

= r x.x. ^ - X, z A. .

^^^ 1 i,n k'^i=i 1

Since X^ has a multivariate normal limiting distribution it

follows (Serf ling, 1980, p. 18) that any linear combination

■
of the X^ must converge in distribution to the same linear

combination of a random variable with that multivariate

normal distribution. Hence

^-1 k-1 k-1 k-1 „ k-1 _

,i,^i^i,n - ^k,n.i/i . N(.Z^X...-,^_Z_^A.,a2r(_z^.2

) +

k-1 ^

(.^ ^i) ]) (3.4.33)

as n ^ °o . Letting

A =

k-1

(3.4,33) can be expressed as follows

154

A'^n t NU'y°,a2A'(Ik_i+Jk_i)i)

(3.4.34)

as n -> ". The result (3.4.32) follows (Serfling, 1980,

p. 18), since (3.4.34) implies that any linear combination

of the elements of X° converges in distribution to the same

linear combination of a variable with the
o 2

N.

k_l (Ji 'Cr (Ik_i+J]^_2) ) distribution

Before proceeding with the next lemma, we establish
some additional notation. Let B^, g* , and co^, i = 1 , . . . ,
k, be as defined earlier and now define

!* =

'^

^k

'k
'k

'k

L J

(3.4.35)

155

where ^^ has k elements. For a vector X v/ith k elements,
let X be the vector of k-1 elem.ents formed by subtracting
the kth element from each of the first k-1 elements of ^.
This notation was used implicitly in the statement of Lemma
3.4.6. For example.

6* -B*
■^k-l ^k

and 03 =

'"l-'^k

'

'^i

"2-^k

0)2

«

•

•

•

•

•

'"k-r^k

'^k-l

^

(3.4.36)

since m, = .

Lemma 3.4.7 ; Let _e*° and u° be as given in (3.4.36),

and let

-1

t^ = [p''('i',<^)I*(F)]"-'(Ij^_^+j^_^).

(3.4.37)

Under H (3.4.17) and conditions A-D ,

-N

N'

-O. d

^x<i* ) :Nk-i(^"^^ 'M

(3.4.38)

as N -> °° .
Proof of lemma;

Recall the definition of _6, and _B* in (3.4.35). From
Lemma 3.4.5 and the independence of g* and ^"^ , , for i 7^ i ' ,

= »^ i .— ^1-..— — f.^--.. ., ,—— ^-- ^ ^■^- ..„ - - , — - ^ , , r -

156

it follows that

N^^d*-!^) J Nj^{k-^,[p2(Y,^)i*(F)]-lT )

as N > CO, under E^. Now we identify N^a^{B*-3^) with X

X — — K — n
(3.4.30) and apply Lemma 3.4.6:

N^x([l*-lk]°) t N(k-V,[P^(^,*)I*(F)](Ij^_^+J^_^)),

*

o „*o

^^k-l))

which, since [^ -_3 ] ° = 3 °, implie

n'^^^CB*") f N{k-V.[P^^.<l>)IMF)](Ij^_^

as N ^ =0, and (3.4.38) has been proved.

«

For later use we note that (3.4.23) together with the

i^ if ie

fact that N'a^(B^-B^) has a normal limiting distribution
(3.4.38) implies

^i = °p(l) (3.4.39)

under H^ and conditions A-D, asN^oo, i = i^ ^ ]^_2
By Theorem 8.3.4 in Graybill (1969, p. 171),

V^ = [P^(^,*)IMF)](I^_^-k-lj^_^). (3.4.40)

157

Leinma 3.4.8 : Under H (3.4.17) and conditions A-D,

Na^(l*°)'lv^(l*°) f xJ_i(p2(^,q))I*(F)k-l Z i.,-Z}h

z
i=l

(3.4.41)

as N ->• ".
Proof of lemma :

From lemma 3.4.7 we have the following;

•'V*°^''ic-i'x"V4v)

as N -> ".

~k

Let X be a random vector with the N,_ (k~''a) , | )
distribution. Then it follows (Serfling, 1980, p. 25) that

Na

X

2(B*°)'|;^(1*°) ^ X'tv^X (3.4.42)

as N -> CO. The distribution of X'ty^X (Serfling, 1980,
p. 128) is

2 ,1 -1 o ' x-l Ov
Xk_l (k u ?v ii^ ) • (3.4.43)

Evaluating the noncentrality parameter in (3.4.43), we have

158

0)° tyV = P^(^,<J>)I*{F) [a)°'lj^_^(.°-k"V'jj,_iiii°]

i=l ^ i=l ^

k
p^(Y,<t>)I*(F) E (co.-;;;)^, (3.4.44)

i=l ^

which completes the proof of Leirnna 3.4.8.

We now give the main result of this section, a theorem
specifying the asymptotic distribution of the proposed test
statistic L (3.4.16) under H,,.

Theorem 3.4.9 ; Under H^^ (3.4.17) and conditions A-D,

L t Xk_i(P^(Y.<!>)IMF)k"\z {i^^-Z)^) (3.4.45)

k
as N -> 00, where L is given by (3.4.16) and oj = Z cj./k.

i=l ^

Proof:

k — I

L = Z (V.-V)^ i

i=l ^

= [ z vl'] - [k -^( Z V.)^]
i=l ^ i=l ^

159

i=l ^ ^ i=l ^ i=l ^ k

= Z V^ - k ( Z V )^ + V^ - 2k~-^V, Z v.. (3.4.46]
i=l i=l ^ ^ ^i=l 1

By Lemma 3.4.2 (see (3.4.21))

l\I = Op(l) (3.4.47)

as N > ", and in (3.4.39) we noted

IV^I = Op(l) (3.4.48)

as N ^ 06, i = 1^ — ^ ]^-l^ Consequently, applying these
results to (3.4.46) yields

k-1 k-1.
L = Z V - k ( Z V )^ + o (1) (3.4.49)

i=l ^ i=l ^ P

as N ^ 00. By Slutzky's Theorem (Randies and Wolfe, 1979,
p. 424) it follows from (3.4.49) that if L has a limiting
distribution as N ^ oo, it is the same as the limiting
distribution of

k-1 k-1.
L^ = Z V - k~'( Z V )^ (3.4.50)

1=1 i=l

160

Leirana 3.4.4 showed the following:

Vj, = p(¥,ct,) [lMF)]\^a^(iB|-B*) + Op(l) (3.4.51)

asN->", i = l, ...,k. Once again appealing to Slutzky's
Theorem, it follows from (3.4.51) and the continuity of the
quadratic form (3.4.50), that the limiting distribution of
L^ (and hence of L) is the same as the limiting distribution
of

2 2 ^"^ 9 1 k-1 o

p ('J',f)I*(F)Na^[ Z (e:f-6*)^ - k~-^( E (6*-3*))^].

^ 1=1 ^ ^ i=l ^ ^

(3.4.52)

Using the matrix notation _3*° (3.4.36) and t^^ (3.4.40), the
expression (3.4.52) is equal to

Na;(l*°)'t;^l*°). (3.4.53)

From Lemma 3.4.8, (3.4.53) has a limiting (N^")

2 2 -1 ^ ?

Xj^_l(p {'V,<p)I*{F)k Z (o) -to)'') distribution so (3.4.45)

i=l

holds and the proof is complete.

The asymptotic distribution of L (3.4.16) under

2

^0" ^1 = ^2 " • • • " ^k ^^ ^k-l^°^' "^^^^ follows from

Theorem 3.4.9 if w^ = 03^ = ... = oi^ = . As used in
(3.2.77), let ^y^_^ ^ be the upper 100 (1-a) percentile of the

161

Xj^_-[^(0) distribution. Then the test of H against H
(3.4.1) that rejects H when

^ - ^k-l,a (3.4.54)

is an approximate level a test. Comparisons of the test
(3.2.77) based on S (3.2.74) with this test will be made in
the next section by calculating the tests' asymptotic

relative efficiency.

3.5 Asymptotic Relative Efficiencies
In Sections 3.1 and 3 . 2 we proposed and developed a
test comparing the slopes of several regression lines with a
standard or control. A competitor to this proposed test was
suggested in Section 3.4. We now compare these two tests by
examining the Pitman asymptotic relative efficiency (PARE)
of the corresponding test statistics.

Recall in Section 3.2, assuming model (3.1.1), the
proposed test of the null hypothesis

-0* ^1 " ^2 " ••• " ^k' (3.5.1)

against the alternative,

-1" '^i ^ ^k ^°^ ^^ least one i, (3.5.2)

162

was based on

S - T't ^T, (3.5.3)

where T is a k-1 dimensional vector with elements

T^(0) = (Na^^^)"^ i:Zx^gSgn(Z^g-Z^^), (3.5.4)

i = 1, ..., k-1. The Statistic T.(0) compares the slope of
the ith line with the slope of the kth line. Statistics
having the form of T. (0) were proposed and discussed at
length in Chapter Two, when comparing the slopes of two
lines. The matrix |~ appearing in (3.5.3) is the inverse
of the asymptotic covariance matrix of T (see (3.2.73)).

In the previous section, we introduced a second
statistic for testing H , based on a statistic proposed by
Sen (1969). Again assuming model (3.1.1), recall the
statistics

V^ = [ Z (x.-x)Ej^ ]/(AN'a ), (3.5.5)

j = l -■ ij

i = 1, ..., k, where R . . is the rank of Y. . among the

observations of the ith line, E_ are the scores (3.4.4) or

ID

(3.4.5), and A is the constant in (3.4.7). Let Bf, denote
the Hodges-Lehmann estimate based on V, and let

163

Vi = V.(Y.-3*x) (3.5.6)

equal the value of V. when using Y. - B^x^ , Y. - B*x
••" '^iN ~ ^k^N' ^^^^ ^^^ statistic based on Sen's work is

^ k -

L = Z (V.-V)^ (3.5.7)

i=l

where V = Z V./k.
i=l ^

Let co^, 012, ..., (jjj^ be given constants with u = 0.
For use in the next result define

-1 '^ _ p

^2 _ _i=l_

^w =2 ' (3.5.8)

0)

-1 ^
where w = k E oj . . We derived the asymptotic distribu-
i=l ^

tions of S and L under the sequence of alternatives.

H^: ^i = ^k ^ a)^/N^k^a^, (3.5.9)

-L -1-f •••^ jC.

164

These asymptotic distributions will be used to arrive at the
PARE of the two statistics.

Recall the test (3.2.77) based on the generalization of
the Sievers-Scholz statistic, S (3.5.3), depends on the
underlying error distribution F through A* = 12A (F) . In
practice we suggested a conservative version of the test
(3.2.77) based on replacing A with A given by (3.2.84).
However for asymptotic efficiency comparisons investigating
the relative merits of the Sievers-Scholz and Sen methods,
it is appropriate to use the test (3.2.77) based on S. The
resulting PARE expressions will be evaluated assuming
different specific error distributions.

Result 3.5.1; Let c^, 1(g), a*, p(^,(j)), and I* (F) be

as defined in (3.5.8), (3.2.13), (3.2.12), (3.4.11), and

(3.4.9), respectively. Assuming conditions I-IV and A-D,

the PARE of S (3.5.3) with respect to L (3.5.7) under H

— N

(3.5.9) is

^2 *

PARE(S,L) = ^^^ W) [1 + 1 - 2A

(l-A*)p^(Y,*)I*(F) (l+(k-2)A*)c^

0)

(3.5.10)

Verification of Result 3.5.1 ; Under H^^ and the assumed
conditions. Theorems 3.2.8 and 3.4.9 give the limiting
distributions of S and L, respectively, as follows:

165

k-1 ^ k-1 ^

c

(l+(k-2)A*) Z cor-A*{ E 0).)
(1-A*) (l+(k-2) A*)

(3.5.11)

k
L f X? 1 (p^(^,<j>)I*(F)k~^ E (to.-u)^). (3.5.12)

^ ^ i=l 1

VJith some algebraic manipulations, the noncentrality
parameter for S in (3.5.11) can be expressed as follows;

2 k *
121 (g) .,-1 „ . -x2,r. 1 - 2A ,- _ -^,
* t^ ^ ((i^--w) J [1 + J 5"]. (3.5.13)

1-A i=l -^ (l+(k-2)A )c

CO

Since both S and L have limiting noncentral chi-squared
distributions under H with equal degrees of freedom, it
follows, as discussed in the verification of Result 2.4.5,
that the PARE(S,L) is given by the ratio of the
noncentrality parameters. Then using the noncentrality
parameters for S and L given in (3.5.13) and (3.5.12),
respectively, the PARE (3.5.10) follows imm.ediately .

The PARE(S,L) (3.5.10) depends on the constants

2
,, w-f '•'/ u, through c (3.5.8). Hence we give bounds

for this PARE that are independent of the to ' s . We first

state and prove a needed lemma.

tii

Lemma 3.5.2 : For arbitrary constants oi^ ,

CJ,

,^, . . . , ^^

satisfying w, = and m . ^ for some i = 1, ..., k-1,
define

166

1
0). = (1/i) s u and (3.5.14)

2 ^ - 2
^i - .^ ((Jj-o)^) (3.5.15)

for i = k-1, k. Then

1,-2
V
-2~ - ^~^- (3.5.16)

^k

Proof of lemma;

First note

""k-l = (k/k-l)^;;;^. (3.5.17)

Then

= 2 _ ^ 2 , -2

^ i=l ^ k

,E^ .2 _ (3,_i,-2_^ ^ (k-l)^2_^ _ ^-2

^k-1 ^ ^^ /k-l)a)^_i - k[;;J (from (3.5.17))

167

= sj_^ + kmj[(k/(k-l)) - 1]

^k-1 + [k;;;^/(k-i)]

, -2
k
—2 = (k-l)[l - (sj_^/sj)]. ■ (3.5.18)

^k

The right hand side of (3.5.18) is maximized when s"^ =0

k-1 '

in which case (3.5.16) follows immediately.

Result 3.5.3: The PARE (S,L) given by (3.5.10) has the
following bounds:

121 (g) * 1 ot2 / ^
■* 2~^ * 1 P^^^ (S,L) < — ^^^ ^?/

(1-A )p" (¥,<(,) I'" (F) ■ ■ - A*p2('^,<^)l*(F)

(3.5.19)

Verification of Result 3.5.3 : Recall the bounds for
A = 12A (F) ,

1/3 < A < 1/2,

applicable under continuous F. Then since 1-2 A* > , it
follows that

*

1 - 2A

* 2 - ° (3.5.20)

(l+(k-2)A )c

0)

■"a'MB>Btww'»iair3i^— f— — niiuM n

168

and so the lower bound in (3.5.19) is obtained by applying
(3.5.20) to the PARE (3.5.10). To obtain the upper bound,
first note that (3.5.16) of Lemma 3.5.2 can be expressed as

— 1 k-1. (3.5.21)

c

Applying (3.5.21) to the PARE (3.5.10) yields

2

PARE (S,£) < ^-i2I__(|)_ ^ k ^^ ^ (3.5.22)

p (Y,(j))I (F) l+(k-2)A

Then since

A* < 1/2 => ^ ^ -X_, (3.5.23)

l+(k-2)A A

the upper bound in (3.5.19) is obtained by applying (3.5.23)
to (3.5.22).

Suppose we assume Wilcoxon scores, 4)(u)=u, 0<u<l.
Then, as derived in Chapter Two, it follows from (2.4.34),
(2.4.35) , and (2.4.39) that

P^('i',*)I*(F) = 12l^(f) . (3.5.24)

Hence we substitute (3.5.24) into (3.5.19) resulting in

t" 'Ifll.BHI' 'lipr^'ir ■ iiil'l j-llMLli.ili

169

i^(g)

(l-A*)I^(f)

< PARE (S,L) <

t2.

£L

— * 2 '
A I^(f)

(3.5.25)

assuming <i> (u) = u. Since A = 12A (F) , values of A may be
obtained from Table 2 4 for several common error distribu-
tions.

Table 25 gives the upper and lower bounds of the
PARE(S,L) for four error distributions, assuming (p (u) = u.
We see that the PARE (S,L) is close to one for the three
distributions with light to moderately heavy tails (uniform,
normal, double exponential) , but has value close to 1/2
under the heavily tailed Cauchy distribution. These results
are essentially the same as the PARE comparisons of the
Sievers-Scholz and Sen statistics in Table 2 for the two
line setting. Thus, as remarked at the beginning of this
chapter, the superiority of Sen's test with respect to

Table 25. Upper and Lower Bounds of PARE(S,L) for Selected
Error Distributions Assuming (u) = u.

Distribution

Uniform

Normal

Double Exponential

Cauchy

Lower Bound

Upper Bound

0.87
0.97
0.74
0.46

0.91
1.04
0.83
0.55

170

the Sievers-Scholz test in terms of PARE is not surprising,
given our results in Chapter Two and the fact that Sen's
test maximizes the efficiency relative to the likelihood
ratio test. Note, however, that assuming (f) (u) = u, the PARE
of Sen's test to the classical least squares theory test
under double exponential errors is 1.50 (Sen, 1969, p. 1676;
Hollander and Wolfe, 1973, p. 64) and the corresponding
bounds of the PARE of the Sievers-Scholz test based on S to
the classical test are 1.11 and 1.24. The point here is
that although the PARE (S,L) favors Sen's test under heavily
tailed distributions, the difference between the tests'
PARES with respect to the least squares theory test is not
impressive. Also, recall from Section 3.3 that the

statistic S has an advantage over L since the form of S

s
allows the construction of exact conditional ta'ts of H. that

are computationally feasible under small sample sizes. The

iterative computations necessary to perform Sen's test

preclude the possibility of computationally feasible, exact

tests based on his statistic. Furthermore, simulation

studies by Lo, Simkin, and Worthley (1978) indicate the

power of Sen's test with respect to the classical least

squares test in the case of three regression lines under

small samples is quite conservative. In view of these

points, the proposed generalization of the Sievers-Scholz

method to the setting of several regression lines has merit

as a robust, computationally simple technique allowing exact

tests that are feasible under small sample sizes.

g*TT * —t tm-- i

CHAPTER FOUR
CONCLUSIONS

We now summarize the conclusions resulting from the
work in Chapter Two and Chapter Three. In Chapter Two we
considered the case of two regression lines. Assuming
common regression constants for the two lines, we proposed
the application of a statistic due to Sievers and Scholz
(2.1.2) to the observed differences Z., j=l, ..., n, given
by (2.1.3). The proposed statistic (2.1.2) employs weights,
^rs' ^^^^ ^^® chosen by the user'. If the regression con-
stants are x^, , x^^, then when using the weights

^^2 ~ ^s ~ ^r' ^^^ null distribution of the proposed statis-
tic depends on the regression constants. An associated
exact confidence interval for the slope difference can be
obtained by calculation of a permutation distribution. We
found this to be feasible with the use of a computer when
the number of regression constants was small, say less than

eight per line. When using the weights a = 1 if x > x .

rs s r'

zero otherwise, the proposed test statistic essentially
reduces to a statistic due to Theil and Sen (2.1.4), com-
puted using the observed differences Z . , j = 1 , . . . , n. In
this case, the null distribution of the proposed statistic
depends on Kendall's tau. An associated exact confidence

171

172

interval for the slope difference can be calculated using

readily available tabled critical values of this distribu-
tion. VJe proposed this procedure when use of the weights

a = X - X was not feasible,
rs s r

Pitman asymptotic relative efficiencies (PAREs) of

these two proposed tests with respect to the tests of

Hollander, Rao and Gore, Sen, and the classical t-test were

computed assuming equal spacing of the regression constants.

Choice of the weights a = x - x or the zero-one weights

^ rs s r ^

does not affect the PARE under equal spacing, so results are
given jointly for these two methods. The PARE of the
proposed method relative to Hollander's method is greater
than one over a wide variety of error distributions. The
PARE of the proposed method with respect to the Rao-Gore
technique is 4/3 irrespective of the underlying error
distribution (subject to certain regularity conditions) .
Simulation results assuming unequal spacing of the regres-
sion constants also favor the proposed methods over the
other two exact procedures (the Hollander and Rao-Gore
methods) .

The PARE of Sen's (1969) test with respect to the
proposed test is greater than or equal to one under most
common error distributions. However, Sen's approach is only
asymptotically distribution-free and requires iterative
techniques. The PAREs and simulation results showed the
classical t-test performs better than the proposed methods
under distributions with light tails. However, the

173

classical test does very poorly when the underlying error
distribution has heavy tails, such as the Cauchy distribu-
tion. For these reasons we feel the proposed methods are
preferred when exact, distribution-free procedures are
desired to make inference about the slope difference,
assuming common regression constants are used for both
lines.

In Chapter Three we considered the case of several
regression lines. Assuming common regression constants, we
defined a set of statistics, each having the form of the
proposed statistic of Chapter Two. Specifically, when
comparing the slopes of k lines, k-1 statistics were con-
sidered. The ith statistic compared the slope of the ith
and kth lines. We proposed a test based on a quadratic form
(3.2.74) in this set of k-1 statistics. Hence this test was
designed to detect alternatives where one or more of the
slopes of the first k-1 lines differ from the kth line.
This is the case where the kth line is considered to be a
standard, or control.

A modification of Sen's (1969) test was constructed as
a potential competitor to the proposed test. Loxver and
upper bounds of the PAREs of the proposed test with respect
to the modification of Sen's test were calculated for
selected error distributions (see Table 25) . Although these
bounds favor Sen's test under the very heavily tailed Cauchy
distribution, they are close to one for the uniform, normal,
and double exponential distributions. We showed that exact

174

conditional tests based on the proposed statistic are
computationally feasible under small sample sizes, a feature
not shared by Sen's statistic. Hence the method proposed in
the setting of several regression lines has merit as a
relatively robust, computationally simple technique allowing
exact tests.

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BIOGRAPHICAL SKETCH

Raymond Richard Daley was born in West Palm Beach,
Florida, on March 3, 1957. He moved to Titusville, Florida,
in 1962. After graduating from Titusville High School in
1975, he enrolled at the University of Central Florida. He
received his Bachelor of Science degree in statistics in
1978 and was honored at his commencement for having attained
entered Graduate School at the University of Florida in 1978
and later received his Master of Statistics degree in 1980.
He expects to receive the degree of Doctor of Philosophy in
December, 1984.

His professional career has included work as a statis-
tician for an Air Force climatology study at the University
of Central Florida and consulting in the Biostatistics Unit
of the J. Hillis Miller Health Center at the University of
University of Florida. He is a member of the American
Statistical Association and the Biometric Society.

178

_ _I certify that I have read this study and that in my
opmxon It conforms to acceptable standards of scholarly
presentation and is fully adequate in scope and quality, as
a dissertation for the degree of Doctor of Philosophv.

\

Pen aver V. Rao, Chairman
Professor of Statistics

I certify that I have read this study and that in my
opinion It conforms to acceptable standards of scholarly
presentation and is fully adequate in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.

U M d?a.

Ronald H. Randies
Professor of Statistics

I certify that I have read this study
opinion it conforms to acceptable standa
presentation and is fully adequate in sc
a dissertation for the degree of Doctor o

and that in my
s of scholarly
and quality, as
f Philosophy.

rd

ope

C^ . CT^ ^

JohnjGp Saw

Professor of Statistics

```